I would distinguish "make any movement" from reacting to and taking evasive action from an errant pitch to protect himself. No way I see this as INT.

That's not what happened in the OP. This was after the pitch.

As I'm reading and it envisioning it, the batter is recovering and standing from his evasive action in the same motion. Not INT to me.

1. Leaving the box has nothing to do with the criteria for judging interference in this play.

2. Did he interfer with the throw? The batter can not vaporize however there was an area that the catcher deemed available to throw the ball and then, up popped a head.

Again it was a HTBT situation and possibly a natural evasive movement which may have negated calling interference but, I still think BI happens way more times than it is called at the amateur level.

That wouldn't be part of his movement in reaction to the pitch.

It certainly could be, depends on how it happened.

No, basically, I did not mention intent, and while I agree that intent has nothing to do with it I see no interference in the batter's actions. It was the fault of the catcher.

On the contrary, whether or not the batter left the box is one of the criteria for judging batter interference. In this situation, because the batter remained in the box, he is afforded a degree of protection.

NFHS Casebook: 7.3.5E: With less than two outs, R1 on second and B2 at the plate, R1 attempts to steal third. In the process, B2, who bats right-handed, after swinging or not swinging at the pitch (a) makes no attempt to get out of the way of F2 throwing to third or (b) is unable to make an attempt to get out of the way of F2 throwing to third. As a result, F2 cannot make a play on the runner. Is B2 out and must R1 return to second?

RULING: B2 is not guilty of interference in (a) or (b). B2 is entitled to his position in the batter’s box and is not subject to being penalized for interference unless he moves to re-establish his position after F2 has received the pitch, which then prevents F2 from attempting to play on a runner. Failing to move so F2 can make a throw is not batter interference.

According to OP, the only movement the batter made after the catcher received the pitch was to stand up in place.

So the question is: Does standing up in place constitute re-establishing position?

IMO, no. We always determine a players position based on the placement of his feet e.g. In/Out of running lane or In/Out of the batter's box. If his feet haven't moved, he hasn't re-established a position in the box.

Personally, I would not penalize the batter in the play described above.

Your right it is "ONE OF" the criteria but, definitely NOT the only deciding factor in making BI determinations. As shown in the Case play. I agree.

As far as re-establishing position, the case play answers this however, you choose to disagree with the ruling. Why the batter sttod up and interfer and the fact that he DID interfer are two different things. INTENT has NO bearing on the call.

As far as feet moving being directly related to re-establishing position, I disagree, unless you have some official authoratative interpretation relevant to that.

Does the ball remain live after hitting the batter/equipment?

There was a play this morning where F2 tried to throw to F3 to catch R2 stealing. The throw hit the batter's bat while he was in the box and ended up going against the 3rd base dugout. R2 ended up scoring and R1 ended up on third.

Coaches were questioning if the play should be declared dead after hitting the bat.

If there's nothing, it's live. If there's interference, it's live until the initial throw doesn't retire the runner being played upon.