Mechanics questions
 

Two umpire system:
Situation A: PU has 3rd, but we have a rundown with PU at HP end and BU at 3rd end.
1) If the tag is made at the 3rd end, is it then the BU call?
2) If there are trailing runner(s), should the BU stay with them or help with rundown?

Situation B: BU customarily or by pregame has the "first infield play",
but it's a throw from the OF to 3rd.
3) Is it then the PU call?
4) Same scenario, what if there are trailing runner(s)?

Situation C: Multiple runners, BU in C.
Batted ball caught in air, thrown to 1st for live ball appeal.
1) Whose primary responsibility?
2) Same scenario, but only runner at 1st, BU in B?

If I can look these up, feel free to tell me I'm just being lazy. The obvious answer is verbal communication between umps at the time, but I'm asking if there are prescribed answers in any sanction.

My opinion in blue.

Situation A: PU has 3rd, but we have a rundown with PU at HP end and BU at 3rd end.
1) If the tag is made at the 3rd end, is it then the BU call?
> That is my understanding.
2) If there are trailing runner(s), should the BU stay with them or help with rundown?
> I would help with run down, remembering that the trailing runners are still my responsibility.

Situation B: BU customarily or by pregame has the "first infield play",
but it's a throw from the OF to 3rd.
3) Is it then the PU call?
> I personnally do not consider a throw from the outfield to be the "first infield play." If I was the PU, I would consider it my call.
4) Same scenario, what if there are trailing runner(s)?
> Same answer (aren't there always going to be trailing runners, i.e. the BR?)

Situation C: Multiple runners, BU in C.
Batted ball caught in air, thrown to 1st for live ball appeal.
1) Whose primary responsibility?
> It depends on how many runners and where they are.
2) Same scenario, but only runner at 1st, BU in B?
> I believe BU has tag responsibilities.

My comments in Blue also:
Two umpire system:
Situation A: PU has 3rd, but we have a rundown with PU at HP end and BU at 3rd end.
1) If the tag is made at the 3rd end, is it then the BU call?
Maybe I am missing something but where does it say in ASA and FED that 3rd base is the PU responsibility in 2 person? Now if you mean PU came up on a long fly to help BU then ok, or lead runner is coming into 3rd ok. Yes if BU is helping, then they should communicate "I got 3" and then there is no doubt and no double calls, 3rd is BU.
2) If there are trailing runner(s), should the BU stay with them or help with rundown? IMO the ball takes you to the play, make sure you look for OBS on trailing runner but get to help your partner as soon as you can, the trailing runners can only go so far with a runner caught in your condition...so as mentioned stay with the rundown understanding that you will have to bust to get an angle if they make a crazy throw to get the trail runner somewhere.

Situation B: BU customarily or by pregame has the "first infield play",
but it's a throw from the OF to 3rd.
3) Is it then the PU call?
depends, is this a play on the lead runner? or the BR? If BR the BU should be taking runner to 3rd. so it would be their call, now if there were runners on 1st and 2nd then it might be the BU's as they would be in position possibly, but I would think as PU I better be in position to make that call if needed, if BU is on it I back down if they are focused on BR then I better call it and I would communicate to my partner as I came in to call it.
4) Same scenario, what if there are trailing runner(s)? I think it is the same as I mentioned above

Situation C: Multiple runners, BU in C.
Batted ball caught in air, thrown to 1st for live ball appeal.
1) Whose primary responsibility?
2) Same scenario, but only runner at 1st, BU in B?
I think BU in both cases.