noumpere

If R1 is tagged BEFORE reaching second, it's a force out and the run does not count.  IF R1 is tagged AFTER reaching and rounding second, it's not a force out and the run counts.

Run counts.  All codes.
 
Here's the OBR:
 
Rule 5.06(b)(3)(B) Comment: A runner forced to advance
without liability to be put out may advance past the base to
which he is entitled only at his peril. If such a runner, forced to
advance, is put out for the third out before a preceding runner,
also forced to advance, touches home plate, the run shall score.
Play. Two out, bases full, batter walks but runner from second
is overzealous and runs past third base toward home and is
tagged out on a throw by the catcher. Even though two are out,
the run would score on the theory that the run was forced home
by the base on balls and that all the runners needed to do was
proceed and touch the next base.

Yes.  Sometimes even before it stops rolling (at least in theory).  And it's true whether the bat is in fair or foul territory.
Yes.  So can the catcher, the umpire, ...
 

It depends a bit on what you mean by "fouls out."  If you mean the better hit a ball that was foul and not caught, see Lou B's answer.
If you mean the ball was a "caught foul fly," then I *think* it's scored as a PO for whoever caught the ball.

To the OP:  Note that tagging a RUNNER is different from tagging a BASE.  The former requires tagging with the ball or the hand holding the ball; the latter just requires touching the base with the defender's "person."

Once the defense tagged second base, R1 was out on the force.  Since R2 is no longer forced, he is safe when tagged while standing on second base.
Had the defense tagged the runner first, he would be out.  then if the defense stepped on second base, R1 would also be out.

The trailing runner is out.  The base belongs to the lead runner until either (a) the lead runner is forced from the base by the batter becoming a batter-runner, or (b) the lead runner acquires the next base.

It's required, but only before re-taking the rubber.  Unfortunately, it's a common myth that failing to do so (nearly) immediately is a balk.

The pitch to B5 "legitimized" B6's at-bat.  B7 *should have been* the next batter, but that's only going to matter if the defense appeals.
 
So far, the defense hasn't appeals, so only B6 is out.  B5 is at first, and B6 is the next batter (the batter after B5).
 
But, if the defense now appeals, B7 is out (for failing to bat after B6), and B8 is the next batter.  B5 is removed from the bast, and two are out.

None.
 
No runs can score when the batter makes the third out before legally reaching first.
 
Here's an AR, right from the OBR rules book:
APPROVED RULING: Two out, bases full, batter hits
home run over fence. Batter, on appeal, is declared out
for missing first base. Three outs. No run counts.

Assuming it was unintentional:
1) OBR:
A) The runner is out.
B) Unless it would be "unfair" to the runner because a fielder had a legitimate attempt at fielding the ball and the ball went immediately through or past the fielder and hit the runner immediately behind the fielder -- in this instance the runner is safe.
C) Unless that exception is unfair to the defense because another fielder had a chance to make a play -- in this instance the runner is out.
2) FED:
A) The runner is out.
B) Unless the ball was beyond all infielders who had a play (sometimes called the "String theory -- draw a line from F3 to F4 to F6 to F5 -- if the ball passes that line, the runner is safe)
 
 
 

It's a time play.  If the appeal /out happened first, the run does not score.  If the run happened first it does score.  It's just that simple.

A run does not count when the batter makes the third out before reaching first base.  It doesn't matter if he's put out by F3 touching first, or by being tagged, or by being appealed that he missed first, or declared out for returning to and touching home (your example, but I'm making some assumptions on what happened).
 
(and, technically, the batter is never forced to first, but it doesn't really matter)
 

The timing on your play is a bit unclear, but, in general, yes -- the force can be reinstated by R1 returning.
 
Here's the OBR rule (emphasis added) -- all codes agree.
 
(6) 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. However, if a following runner
is put out on a force play, the force is removed and the
runner must be tagged to be put out. The force is removed
as soon as the runner touches the base to which he is
forced to advance, and if he overslides or overruns the
base, the runner must be tagged to be put out. However, if
the forced runner, after touching the next base, retreats
for any reason towards the base he had last occupied, the
force play is reinstated, and he can again be put out if the
defense tags the base to which he is forced;

The runner is out for INT.  The fielder's protection incluses while he is preapring to field the ball, while he is fielding it, and through making a throw following it.
 
This is from some version of MLBUM:
Note that under the Official Baseball Rules, a fielder is protected while in the act of fielding a
batted ball. In addition, a fielder is also protected while in the act of making a play after having
fielded a batted ball. If, after a player has fielded a batted ball but before he is able to throw the
ball, a runner hinders or impedes such fielder, the runner shall be called out for interference.
Furthermore, a runner who is judged to have hindered a fielder who is attempting to make a play
on a batted ball is out whether the hindrance was intentional or not.

Gives the pitcher a ball, puts it in play (even without a batter), and has the defense execute an appeal.

These answers, combined, are correct.  imo, of course.

Correct by rule; correct by the CS&FP handbook.

That's the OBR and NCAA rule.  The CURRENT FED rule does not allow the "overrun protection" on a walk.  (Although, most umpires will allow it if it's D3/K or B4 situation).  Starting next year, the FED rule will be (at least substantially) the same as the OBR and NCAA rule.

Well, it *is* throwing to an unoccupied base -- it's just that this is allowed if there is a play being made. 
 

The run counts -- for now.  The play at second is a time play, and since the run crossed before the appeal was made, the run scores (the "time" for scoring the run is the time the appeal is made -- not the time either runner leaves the base).
 
But, you could still appeal at third to get an "advantageous fourth out" that would negate the run (assuming the infielders are still in fair territory)

Absent the INT, would there have been a double play?  In your pop-up scenario -- no.  BR would be out and R1 would return to first.  Since there was INT, change that to R1 is out and BR is awarded first.
 
Note that if this was an Infield Fly situation, that the BR is out on the infield fly, AND R1 is out for the INT.
 
(FED has a different standard for getting two outs here -- but it still wouldn't apply on the pop-up situation.)

No.  But, enforce it as if it does at the levels most of us do.

Yes, ti counts.  R2 was not forced TO second, so it's not a force out.  It's a time play -- and since the run scored before the out occured, the run counts.

See #5 below (the rules book, including the scoring, is available on line):
9.05 Base Hits
A base hit is a statistic credited to a batter when such batter reaches
base safely, as set forth in this Rule 9.05 (Rule 10.05).
(a) The official scorer shall credit a batter with a base hit when:
(1) the batter reaches first base (or any succeeding base)
safely on a fair ball that settles on the ground, that
touches a fence before being touched by a fielder or that
clears a fence;
(2) the batter reaches first base safely on a fair ball hit with
such force, or so slowly, that any fielder attempting to
make a play with the ball has no opportunity to do so;
Rule 9.04(a) to 9.05(a)

Rule 9.05(a)(2) Comment: The official scorer shall credit a hit
if the fielder attempting to handle the ball cannot make a play,
even if such fielder deflects the ball from or cuts off another
fielder who could have put out a runner.
(3) the batter reaches first base safely on a fair ball that takes
an unnatural bounce so that a fielder cannot handle it with
ordinary effort, or that touches the pitcher’s plate or any
base (including home plate) before being touched by a
fielder and bounces so that a fielder cannot handle the
ball with ordinary effort;
(4) the batter reaches first base safely on a fair ball that has
not been touched by a fielder and that is in fair territory
when the ball reaches the outfield, unless in the scorer’s
judgment the ball could have been handled with ordinary
effort;
(5) a fair ball that has not been touched by a fielder touches a
runner or an umpire, unless a runner is called out for having
been touched by an Infield Fly, in which case the official
scorer shall not score a hit; or