maven

Last question first: any umpire who sees INT may call it. Since F6 did not throw the ball, the BU could well have had a great look at it.
Second, the timing of the BR touching 1B is irrelevant: the issue concerns the retired runner, R1, and his actions around 2B.
I believe Babe Ruth plays a modified version of pro rules (OBR). The pro rule (6.01(a)(5)) states that a retired runner is guilty of INT if he hinders the defense playing on another runner. The runner is not required to slide, but if he goes in standing he's liable to be called for INT. The proper penalty for his INT is to call out the other runner likely to be played on, in this case the BR.
We can't assess the umpire's judgment without video. But if R1 did alter the play of F6 — say, by making him choose between throwing into R1 or not throwing — then I would say the ruling was correct, and the penalty applied was also correct.
R2, R3 should have been returned to their bases due to their teammate's INT.

If ruled out on the IFF, the BR is as out as out can be. His status as a retired runner doesn't depend on an umpire instructing him to return to the dugout. That instruction can be given at any subsequent time.

Yes. R2 is not forced to advance, so this is a time play. As the run scored before the third out was made, it counts. Same in all codes.

No, it's a force play whether he tags the runner or the base.

In general, BU must first assess whether this is a trouble ball. A "can of corn" is the opposite of a trouble ball. So BU should not go out on a can of corn.
When BU comes in and pivots, he has primary responsibility for the BR, the touch of 1B, and action around 1B. PU has primary catch/no catch responsibility.
BU has secondary responsibility for catch/no catch if he has nothing better to look at.

No. It's a force play on R2 for the third out. A force out can be recorded by tagging the runner or the base to which he is forced to advance by the batter becoming a runner.

No rule dictates proper signals. Signals are part of umpire mechanics, and proper signals are given in mechanics manuals.
The proper mechanic for "no catch" is to signal "safe" and verbalize "no catch." There's no special "version" of the safe signal for "no catch," and pointing to the ground is a not standard signal.
Mechanics such as signals are tools for communicating our calls. Where no communication is needed — a clean base hit bouncing 3 times in front of an outfielder — no signal is needed. On a very close "trap" play, both signal and verbalization should be used. For plays in between the obvious and the very tight, we can use just a signal. When in doubt, use both.

The rule-book zone is not a 2-dimensional rectangle above the front of the plate (as seen on TV), but a 3-dimensional volume over the entire plate. A pitch is a strike by rule if any part of the ball touches any part of this imaginary volume above the plate. Where F2 catches a pitch provides information about where it was when it was over the plate, but does not tell us by itself whether a pitch was a strike.
In practice, many umpires call a ovoid-shaped zone, which has an oval profile instead of a rectangular one. For that kind of zone, a pitcher can "miss" in 1 dimension and still get a strike: a little up or down and over the heart of the plate is a strike, as is a little in or out and thigh high. But "up AND in" or "down AND away," and that's a ball.
I give more east-west than I do north-south (which is easier for the dugouts to see and complain about).

Assuming that there was 1 out before all the fun and games started, the batter made the second out immediately on the strikeout. If R3 scored before R1 was tagged out for the third out, then the run counts. If not, then not.

I'm also guessing that this has to be FED.
If so, part of why it doesn't seem right is that people import assumptions about the DH and substitutions from other codes. Just remember:
The DH and the fielder he bats for share a lineup spot: one plays offense, the other plays defense. Both are starters and have reentry privileges. As long as each is doing his job, they're both in the game. When one does the other's job, the role of the DH is done, and the other is out of the game (and may reenter). We don't care where fielders play on defense: if they're in the game, they're in their proper lineup spot. So in your play, it's easy to see that the substitutions are legal. P2/P3 share a lineup spot. When P3 plays defense, he takes over his lineup spot, that he had been sharing with P2, who is now out of the game (may reenter).
When P2 reenters, he occupies his prior lineup spot, and P3 must leave the game (and he may still reenter). P1 is still in the game unless substituted for separately.
If you have P2 and P3 both on the field on defense, then you have an illegal substitution situation: they occupy the same lineup spot, and may not be in the game at the same time.

That's correct. So to the OP: can A remain as DH? Simple answer = no.
Longer answer:
In the OP, A is DH, then enters on defense to pitch. That ends the role of the DH, and it sends starter B, who was pitching, out of the game. Now A is occupying both halves of his lineup spot, offense and defense. B has his re-entry privilege.
When C enters to pitch for A, now A is out of the game, and C occupies his lineup spot. A has his re-entry privilege.
Both A and B may re-enter the game subsequently, and (provided no pitching provisions have been violated) either one may also pitch. They may not enter at the same time, as they will have to occupy 1 line of the lineup.
If coach wanted later to enter A to "bat for" C, as if he were still DH, then he may do so as a simple substitution (not as DH). However C, as a substitute, has no re-entry privilege, and subbing for him on offense would mean he's done for the day.

This term engenders some confusion: "the base path" is determined by the runner, not any lines on the field. It's a straight line from wherever he is to the base he's running toward.
When a fielder tries to tag him (not near a base), he can legally do 3 things: (1) keep running toward the base he's trying to attain (advancing or retreating) by running 3 feet or less from the path he was on, or (2) turn around and head to the other nearby base (advance or retreat, and this move creates a new base path), or (3) give himself up.

You are correct: tagged in that order (first R3, then 3B), the result is a double play.
You want OBR 5.09(b)(6)/7.08(e): "Any runner is out when...he or the next base is tagged before he touches the next base, after he has been forced to advance by reason of the batter becoming a runner." There is no exception for a runner standing on the base to which he was formerly entitled.
This rule covers both R2 and R3: when the batter became a runner, R3 was forced to advance, so when he was tagged he was out. R2 was also forced to advance, and when 3B was tagged, he was out.
Had the fielder first tagged 3B, that would have put R2 out and removed the force on R3. R3 would then be entitled to 3B again and would not be out when tagged.

As described — or at least as I'm envisioning it — that's not a correct call: being out of the baseline is not an infraction. Being out of the baseline in order to avoid a tag is. And your runner was changing directions and making a new base path, not avoiding a tag by going out of one.
The rule prohibits running around a fielder with the ball trying to tag the runner. The runner may dodge, duck, or otherwise avoid a tag, but running wide of the fielder with the ball is prohibited, and yields a live-ball out.
Notice this provision has nothing to do with a runner's "normal path," whatever that might be. A runner's path is between him and the base he's trying to acquire (forward or backward). If he changes directions, he has a new path.
If there's a teachable moment for your runner, I suppose it's that sometimes the breaks don't go your way. You found an umpire who blew a rule. If you let luck — good or bad — get in your head, then you're not sufficiently focused to excel in the game.

Name. This question typically arises in connection with batting out of order: the lineup goes by player, not number. If the numbers are incorrect, they may be fixed without penalty.

No. 6-2-4a

Let's look at the rule.
 

So, there are two main ways a right-handed batter is guilty of BI on a throw to 3B:
He steps out of the box AND hinders F2 He remains in the box AND makes some "other movement" AND hinders F2 Note that hindrance alone is not sufficient for BI: the batter must do something else, either stepping out of the box (1 or 2 feet), or some "other movement."
 
So, if you're coaching batters, the best thing to teach them is to stay in the box and not make any other movement (ducking, turning, etc.). Any hindrance in that case should be ruled incidental, since the batter is not expected to disappear.
 
A batter who swings at the pitch is permitted to complete his swing normally, and that would not count as an "other movement" for the purpose of this rule. Swinging and falling into F2's way, though, would, and should be ruled BI.
 
The FED rule (7-3-5) is substantially the same and gives more explicit guidance about what constitutes BI (leaning over the plate, etc.).

Legal. Assuming only that he steps toward 2B, he is throwing to an unoccupied base for the purpose of making a play. OBR 8.05(d), NFHS 6-2-4

Fly ball touched while ball is over foul = foul ball. Caught or uncaught, in front of the base or beyond, nothing else matters. 2.00 FOUL BALL

Judge whether it's a fly ball. If so, foul out. If not, foul ball. OBR 2.00 FLY BALL, FOUL BALL, CATCH, IN FLIGHT Catch, batter is out. OBR 2.00 CATCH, IN FLIGHT

OBR: not legal. Once removed from the game, any player is done for the remainder of the game.
 
FED: You don't mention whether these moves involved (a) substitutions, or (b) just players changing positions on defense. (a) Returning the starting pitcher after substituting for him is legal, provided his re-entry privilege is intact (some pitching infractions can negate it). Returning the substitute pitcher to the game after he has been removed is not legal, since substitutes do not have the re-entry privilege. (b) If both pitchers were in the lineup and simply changed positions, then the moves listed are legal: a pitcher may be removed and returned to the mound once per inning, and the team did that with each pitcher as described.
 
LOCAL: who knows? I don't see why a continuous batting order would be relevant, since that is not a substitution rule for the defense.
 
In none of these cases would violations result in a forfeit, which is the nuclear option of penalties.

No, we're fairly strict about what counts as stepping off or disengaging: it must be a step directly backward and off the rubber with the pivot foot.
 
Technically, the disengaging foot must touch the ground before the hands separate in a move toward 1B, but we're not as strict about that. Properly coached runners know to get back once the pivot moves.
 
If he makes that pivot move and throws the ball away, we're awarding 1 base, not 2, because he was still engaged. That same ruling makes feinting a balk.

7.09(e):
"Any batter or runner who has just been put out, or any runner who has just scored,
hinders or impedes any following play being made on a runner. Such runner shall
be declared out for the interference of his teammate."

Batter INT. The batter may not step out of the batter's box nor make any other motion (besides swinging at the pitch) that hinders the catcher's play on a runner. Intent is irrelevant.
 
The ball is dead, the batter is out, and R2 must return to 2B. If the batter struck out before interfering, then R2 is out for the INT. 6.06(c )
 
Edited to add: plus ice for the batter's neck. Ouch!

Next player in lineup. His time at bat ended on ball 4.