A pitch is considered a pitch until one of the following things happens (NFHS 2-28-4)

1. The ball is secured by the catcher
2. The ball is comes to rest
3. The ball goes out of play
4. The ball becomes dead
5. the batter hits the ball (other than a foul tip)

A pitch that bounces in the dirt is still a pitch until one of the items listed above occurs. If it hits the batter, it is still considered a pitch. The ball becomes dead as soon as it touches the batter, and the batter is awarded first base.

A ball that bounces and then goes through the strike zone is a ball. A strike is defined as a pitch that passes through the strike zone "in flight" Once it touches the ground, it is not "in flight", but it is still a "pitch" until one of the items listed in the previous response occurs.

The batter can swing at any pitch, including one that bounces in the dirt . If he swings at it, it is treated the same as any other pitch at which the batter swings.

For purposes of this discussion, the runners will be defined R1 on first and R3 third.

All rules (NFSH, NCAA, OBR) state that a runner acquires the right to the next unoccupied base if he touches it before he is put out. The runner is then entitled to that base until he is put out, he touches the next base while it is unoccupied, or until he forced to advance.

The key to this situation is to determine which base R1 legally occupied at the "time of the pitch". From the set position, the time of the pitch is defined as any motion, after the pitcher has come set, that starts his delivery.

Your scenario indicates that R1 had acquired 2nd base before the pitcher came set. If that is the case, R1 is now legally on 2nd base at the time of the pitch and is only required to tag up at 2nd base on the caught line drive. He would not be out on the appeal at first base.

If, on the other hand, R1 had not reached 2nd base at the "time of the pitch" then R1 would have to return to 1st base and tag up on the caught line drive. If that were the case, R1 would be out on the appeal.

This situation is governed by NFHS Rules 5-1-2-a-1 and 7-3-5 (Penalty),

One of the important considerations in this play is the fact that the runner is attempting to steal third base and not home. As soon as the ball got past the third baseman the ball is dead immediately, since the runner was not put out at the end of the catcher's throw (see Rule 5-1-2-a-1).

Since the runner was not put out on the throw, the batter is out and the runner returns to second base, unless the batter's out is the third out of the inning (see Rule 7-3-5 - Penalty).

If the runner had been attempting to steal home, and there were less than two outs, then the runner would be out instead of the batter. However, in this circumstance, the runner's attempt to advance to home was because of an errant throw caused by batter interference. The ball became dead as soon as an out was not made at third base.

NOTE:
Umpires know that batter's intereference is a delayed dead ball, but many of them do not understand, or know, that once the catcher's throw is completed and the runner is not put out as a result of that throw, the ball is to be called dead immediately. Most let everything continue and then rule after all playing action is over believing that is the correct way to handle the play.

This is a legal play and is known as "Skunk in the Outfield".  The explanation is rather lengthy.  CLICK HERE to see the full explanation of this play and how to umpire it

Under NFHS, when a batter is hit by a pitch, the ball is immediately dead (5-1-1) and the batter is awarded first base (8-1-1-d).   When a ball becomes dead, each runner must return to the base legally occupied at the time the ball became dead (8-3-9).  When a ball becomes dead, a runner may advance only when awarded a base(s) for an act which occurred before the ball became dead (5-2-2-c).

In this situation, the batter would be awarded first base and the runner returned to second base.

Under all rules, if a batter is touched by a fair batted ball before it has passed any infielder (not counting the pitcher), the ball is dead immediately and the runner is out.  The batter is awarded first base and all other runners return to the base occupied at the time of the pitch, unless forced by the batter-runner being awarded first base.

In your situation, no runners would score.  The runner from 1st base is out on the interference, and the batter gets first base.  If the ball had hit a different runner, say the runner from second, then that runner would be out and all runners return to their bases.  However, the batter would be awarded first base, and the runner who is on first would be "forced" to move to second base.  In any circumstance, under this scenario, you would end up with an out, no runs scoring, and bases loaded at the end of all base awards.

The term "delayed dead ball" is a valid term under all rules.  It refers to an action that causes a possible base award or other penalty at the end of playing action and the ball does not become dead until that time, regardless of when the infraction occurred.  Such cases include catcher interference, batter interference, balk (under NCAA and OBR only), and obstruction.  In those cases, the umpire must make note of the infraction and when all playing action has concluded, determine if the penalty is to be enforced.

In NFHS there is only one type of obstruction on a runner, and it is always a delayed dead ball.  It does not make any difference if a play was being made or not. 

In NFHS, a coach is allowed three "charged" conferences during the regular inning (first 7 innings) to the mound without penalty.  He can take them at any time during the regular innings of the game, including to the same pitcher with the same batter.  There is no restriction on him going to the mound with the same batter.  However, if the coach has already used his three conferences, he must remove the pitcher on any subsequent visits.  It doesn't matter if that is the first visit with that pitcher, or the second.  If he is out of "free" conferences, the pitcher must be replaced.  The only exception is if the visit to the mound would violate the pitching restrictions. 

For example, the coach has already used all of his "free" conferences.  He just replaced Pitcher A with Pitcher B.  Pitcher B is pitching to his first batter and is struggling and the coach wants to go out to the mound.  In this case, the coach would not be allowed to go to the mound since his going would require him to replace Pitcher B, and that would be a violation of the pitching rules which require Pitcher B to pitch to the first batter until he is put out, reaches first base, or the third out of the inning is made.

The batter striking the catcher's mitt during attempt to hit the ball is catcher's "interference".  It is, under all rules (OBR/NCAA/NFHS), a delayed dead ball.  The umpires let play continue.  At the end of all playing action, the umpire will call TIME and allow the defensive coach the option of taking the result of the play or enforcing the penalty for catcher interference.

If the coach opts to take the result of the play, the batter remains at bat and the runners are wherever they ended up at the end of playing action.  Any outs made on the play stand and any runs made count. For example, let's say the catcher tries throws to third and the ball goes over the fielder's head deep into the outfield.  The runner from second scores and the runner from first ends up on third base.  The coach must decide if the run and a runner on third are more important to him than having the bases loaded, but without the run scoring.

If the coach opts to enforce the catcher interference, then all outs and runs scored are nullified.  All runners return to the bases occupied at the time of the pitch.  The batter is awarded first base and any runners forced to advance due to the batter being awarded first base, are also awarded the base to which they must advance.  In this case, you now have the bases loaded.

Under NFHS, batters hit by a pitch are governed by rules 5-1-1-1(a), 5-2-2 8-1-1(d), 8-1-1-(d)(1).  If a batter swings at a pitch and is also struck by the pitch, the ball is dead immediately and the pitch is called a strike. (dead ball strike).

DO RUNNERS ADVANCE?
NO.  Runners may not advance on a dead ball unless they are awarded bases or are forced to advance because of the batter being awarded first.  Since the batter was struck by a pitch, only the batter can be awarded a base. (8-1-1 and 8-1-2).  It doesn't make any difference if he swung at the pitch for purposes of base awards.  If he had not swung at the pitch, would you award a base to runners who were not forced? No.  The same applies if he did swing at the pitch. 

DOES BATTER GO TO 1st?
NO.  The batter swung at the pitch and it is a strike.  The base award to the batter is ignored if the pitch is called a strike (8-1-1-d-1), but the ball is dead.

DROPPED THIRD STRIKE?
NO.  The ball is immediately dead as soon as it touches the batter.  Once the ball is dead, it becomes irrelevant if the catcher is able to catch the ball.  No runners may advance and the pitch is called a strike.  If it is a third strike, the batter is out since no runners, including the batter-runner may advance when the ball is dead and the defense cannot make any play that would cause a runner to be put out. (5-2-2)

It is legal under all rules EXCEPT NFHS to make a pickoff from the windup position.  Very few pitchers do it since it is difficult to do without getting balked, but is legal under OBR and NCAA rules.

As an umpire, you have to pay close attention to the foot and make sure it goes directly toward the base and not in his "normal" step that starts his pitching motion.  This is critical since it is now permissible to take a step sideways in order to start the pitching motion.  MLB and MILB recommend that you look at the toes of the non-pivot foot.  If it is facing more toward the plate than the base, it is a balk.  If it is pointed more toward the base, then it is a pickoff move.

Your question concerns whether or not a base coach who is struck by a thrown ball has interfered with the play or not.  Part of this will depend on which set of rules are being used for the game as there is a slight difference between MLB/OBR and High School/NCAA rules.

The rules that govern this situation are as follows:

MLB/OBR - Rule 5.08
      "If a thrown ball accidentally touches a base coach, or a pitched or thrown ball touches an umpire, the ball is alive and in play.  However, if the coach interferes with a thrown ball, the runner is out."

      This will be umpire judgment as to whether or not the coach's actions were accidental or intentional. If ruled intentional, the ball is dead immediately and runner is out.  If accidental, the ball is alive, runners advance at their own risk, and all plays stand on their own merits.

High School - Rule 3-2-3
      "If a thrown ball accidentally touches a base coach in foul territory, or a pitched or thrown ball touches an umpire,  the ball is alive and in play.  However, if the coach is judged by the umpire to have interfered intentionally with     the thrown ball, or interferes in fair territory, the interference penalty is invoked"

      Basically this mean that if the coach is in foul territory and does not intentionally interfere with the thrown ball, the ball is alive and runners advance at their own risk.  If the coach intentionally interferes with a thrown ball while in foul territory, the ball is immediately dead and the runner is out.  If the coach is struck by a thrown ball and is in fair territory, it is automatically interference, the ball is dead, and the runner is out.

NCAA - Rule 8-3j
      "If a thrown ball strikes a base coach on foul ground, the ball is in play. If the coach interfered intentionally with such a thrown ball, the runner is out and any other runners must return to the last base touched"

      This is the same as High School.

The bottom line is that it is umpire judgment, which is not arguable by a coach under any rules, as to whether the coach's actions were intentional or accidental.  In both scenarios you provided, if the touch was accidental, the ball is alive and all runners may advance at their own risk and are subject to being put out if they are not legally touching a base. 

In your first scenario, the batter-runner remains on second base and he acquired it legally.

In your second scenario, the runner would be out as the first baseman touched the base while holding the ball securely in his glove before the batter-runner touched the base. 

If the touch by the coach was deemed to be intentional, then the ball would have been immediately dead, and the batter-runner would be out.  All other runners would return to the bases legally occupied at the time of the interference.

I checked the latest version of the OBR concerning the pitcher’s glove.  You related that the glove was black with grey or white thread/piping along the fingers.  Rule 1.15 addresses this issue and there was a change in 2007-08 which relaxed the rule somewhat under OBR.  I have included the rule below for your reference.  The important change is in section (a) which excludes the piping on a glove from the color restrictions.  From what you described, the glove is legal under the current OBR rules.  NOTE:  If you have a 2007 rulebook, it does not have this change in it since the MLB made the change after the book had already gone to print.   It should be in the 2008 version or you can view the current rulebook on the MLB website at http://mlb.mlb.com/mlb/official_info/official_rules/foreword.jsp.  Even if the glove was deemed to violate the rule, the only penalty is to remove the glove from the game. 

OBR – Rule 1.15
(a) The pitcher’s glove may not, exclusive of piping, be white, gray, nor, in the judgment of an umpire, distracting in any manner.
(b) No pitcher shall attach to his glove any foreign material of a color different from the glove.
(c) The umpire-in-chief shall cause a glove that violates Rules 1.15(a) or 1.15(b) to be removed from the game, either on his own initiative, at the recommendation of another umpire or upon complaint of the opposing manager that the umpire-in-chief agrees has merit.

Texas high school baseball is governed by National Federation of High Schools (NFHS) rules.   NFHS Rule 8-4-1(g) states that the batter-runner is out if he runs outside the three-foot running lane, which is the last half of the distance between home plate and first base) while the ball is being fielded or thrown to the first baseman and interferes with the fielder or throw.  The runner is considered outside the running lane if either foot is outside of either line.

A batter-runner who advances to first base as a result of a hit or an uncaught third strike must remain in the running lane.  In this case, the runner would be out of the running lane since one foot is outside of the lane. When the runner is hit by the catcher's throw, he is guilty of interference by virtue of being out of the running lane and is out.

Yes, that is legal as long as the pitcher is a starting player.  Under NFHS, a starting player may be removed from the game and re-entered one time.  Courtesy runners are not considered “substitutions” under NFHS.  Thus, in this situation, the starting pitcher was removed from the game when a pitch hitter took his place in the batter’s box.  The starting pitcher has been re-entered when he took the pinch runner’s place on the bases.  Now the pitcher is on the bases, and a courtesy runner can be used. 

NOTE:  the courtesy runner could not be used for the pinch runner since he is not the pitcher.  That would have been a substitution.  By re-entering the pitcher to run, the coach now has the option of using a courtesy runner.

Texas High Schools play under the Nation Federation of High Schools (NFHS) rules for baseball.  The short answer is no, there is not a mandatory slide rule at the plate in high school baseball.

 The rules for sliding are covered in rule 8-4-2.  Under this rule a runner is never required to slide, but if he does slide it must be a legal slide.

Under OBR and NFHS it is technically (i.e.: by the exact letter of the rule) a balk.  Whether or not it is called, is going to be based upon the umpire's judgment as to exactly what happened.

OBR (Major League rules) - 
Rule 8.01(b) states that   "Preparatory to coming to a set position, the pitcher shall have one hand on his side; from this position he shall go to his set position as defined in Rule 8.01(b) without interruption and in one continuous motion. The pitcher, following his stretch, must (a) hold the ball in both hands in front of his body and (b) come to a complete stop."   

The issue here is whether the pitcher started to come set when he reached in the glove to get the ball, which will be based upon umpire judgment and will probably be based on several factors such as the length of time between when he stepped on the rubber and reached for the ball, whether both of his feet were in his "normal" Set position, whether the move was intentional and thus designed to deceive the runner, etc.  (see Approved Ruling below)

APPROVED RULING:  "Umpires should bear in mind that the purpose of the balk rule is to prevent the pitcher form deliberately deceiving the base runner.  If there is doubt in the umpires mind, the "intent" of the pitcher should govern."

APPROVED RULING: "A pitcher is allowed to momentarily adjust the ball, or transfer the ball from hand to glove (or vice versa) as he steps onto the rubber, but is otherwise prohibited from from stepping onto the rubber with his hands joined. "

NFHS (High School) -

Rule 6.1.2 governs the Set position.  NFHS also has a book of case interpretations which provide guidance for enforcing these rules. CASE PLAY 6.1.2 SITUATION E states “With R1 on first, F1 receives the ball and, with his feet in set position and in contact with the pitcher's plate, removes the ball from his glove.  This is a balk.  Restrictions on F1's movement begin when he intentionally contacts the pitcher's plate with his pivot foot."

Although this may be technically a balk, whether it is called or the pitcher is warned (by the umpire or by using another player such as the shortstop) is going to be umpire judgment and should be based upon the "intent" of the pitcher and the level of ball being played.  The intent of the balk rule is to prevent deliberate deception by the pitcher.