grayhawk

Yes. You are still required to retouch when the ball is dead and you have been awarded bases. 
 

What you described is a balk.  Once the pitcher is engaged, he can pitch, step and throw to a base, or legally disengage by stepping backwards off the rubber.  The instant he stepped forward with his pivot foot, he has balked.

This should not happen.  When the obstruction occurs, point and verbalize, "that's obstruction!" and allow the play to continue.  When the play is over, you then apply your judgment as to where the runner should be placed.  NEVER EVER verbalize runner placement while the play is live.

The ball is dead and a strike is added to the batter's count.  If he already had two strikes, he would be called out.  In no case would the batter be awarded first.

Foul.  For it to be a foul tip, it must go sharp and direct from the bat to the catcher's glove or hand and then be legally caught by the catcher.  This ball went sharp and direct to the batter's foot - foul ball.

OBR and NCAA:  Legal play as long as he throws.
NFHS:  Balk.  R1 to second.

Unless the runner did something intentional like swat or kick at the glove or the ball, this is nothing.

It's unclear to me where the catcher was when he received the pitch, but it sounds like he was on or over home plate when he caught it.  If that's the case, then catcher's interference could have been called, based on umpire judgment.
FED:  Catcher's obstruction - "when a catcher or fielder hinders a batter"
NCAA:  Catcher's interference - "if any defensive player interferes with the batter's swing or prevents the individual from striking at a pitched ball"
OBR:  Catcher's interference - "The batter becomes a runner and is entitled to first base without liability to be put out (provided he advances to and touches first base) when the catcher or any fielder interferes with him."
Notice that none of the codes require the batter to actually swing and contact the catcher or his glove.  If the catcher "intercepted" the pitch, which hindered the batter, prevented him from striking at the pitch, or interfered with him, then interference (obstruction in Fed) would be warranted.

No.  R1 is out and the BR is awarded first.  R2 and R3 return.
There are situations where the runner getting hit with the batted ball is nothing, and there are circumstances where this would be a double play, but in an effort to give a simple answer to your simple question, I will leave those out.

The umpire didn't take anything away, he merely enforced the rule as written.  The ball is dead the instant it hit the runner in this situation - there is simply no way around that.  Since it wasn't judged to be intentional, R1 is out and the BR is awarded first.  Them's the breaks.

It's not a time play.  Even if R2 scored prior to the appeal, the run would not count because the 3rd out was made by the batter-runner before touching first base.

If he's got a foot out of the box and is hit by the catcher's throw, it is almost always going to be interference.  STAY IN THE BOX!
 
Even if he has BOTH feet in the box, he can still be called for interference if he makes "any other movement" and hinders the catcher's throw or attempt to throw.  The ONLY guarantee to not be called for interference is to stay in the box and NOT MOVE!  Have situational awareness and know what's going on.  Don't do the catcher any favors and try to get out of his way.  Stay put and keep still.
 
The above is for plays where the catcher is making a throw to put out a runner at another base.  In cases where there is a play at the plate and the batter has an opportunity to vacate the area (such as on a wild pitch or passed ball), then the runner must do so or be in jeopardy of being called for interference.  In cases where there is a steal of home, the runner is not expected to vacate because he doesn't have time.

By retreating back toward second base, he reinstated the force, so he does not need to be tagged to be put out. 7.08(e).

The coach was correct.  Players are allowed to dive back to a base, but if they are advancing and slide head first, they should be called out.  The fact that he had already touched that base is irrelevant.  The runner was advancing.

OBR:  The batter must make an attempt to avoid being touched by the ball.
 
Fed:  The batter cannot permit the ball to touch him.

In Fed, it's definitely legal, even if R1 fakes an advance.
 
In OBR, it's more tricky and is up to umpire judgment as to whether or not R1 was actually advancing, or just faking his advance.  In OBR, if the runner was going and turned around to go back to first after seeing F1 use the inside move, then in my judgment he was stealing and F1's move is legal.  If R1 was just taking an aggressive secondary and F1 uses the inside move to throw or fake to second, then it's a balk.

There is NO CHOICE for the coach in this situation.  Either you enforce the balk, or you disregard it.  The ONLY way it can be disregarded is if all runners, including the batter-runner, advance at least one base.
 
Since the batter-runner was put out, enforce the balk, score R3 and bring the batter back to bat with the same count he had before the balk.
 

What rule set?
 
In straight OBR, a feint to third is not legal from either the set or the windup (it was legal prior to 2013).    Some organizations have adopted this rule and others have not.  For example, it's still legal in Little League.
 
In NFHS (Fed) rules, a feint to third is still legal, though you can't pick to any base from the windup.

Doesn't matter where he's headed. This is type A obstruction which is an immediate dead ball and automatic minimum one base award from the last base the runner occupied. Award him home.

1-3-2d:  ...such bats SHALL be labeled with a silkscreen or other permanent certification mark. (my emphasis)
 
This is an illegal bat, and since it was brought to the attention of the umpire before a pitch was thrown to the following batter, the penalty is batter is out and runners return.  Since this was the third out, then the game is over.
 
No need to restrict the coach as long as you can run faster than him to your car...

10.2.3 Situation G.
 
Even though the umpires failed to call the infield fly, both teams are responsible for knowing when it's in effect.  Based on this case play, I would score R3 and keep R2 at third.  The batter should be called out.

R1 to third in all codes for both a and b. F3 touching it or not is irrelevant.


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Tricky.  Seems to me that this is a legal straight substitution.  Both are starters and can be removed once and re-enter the game:
 
Jones was substituted for by Smith, which terminates the DH
Smith was substituted for by Jones
Jones was substituted for a second time by Smith (Jones is now out for the remainder of the game)
Smith is now the pitcher when they take the field, unless legally substituted for by another player
 
That's my best guess.

None.  Third out was a force out, so no runs can score.  BR being called out on appeal does not remove the force on R1.  He was forced at the time of his miss.

Sounds like a double play to me.  A ball that goes up to the batter's head is not sharp and direct.