.3 Foul Question
 

.3 seconds left. A1 catches the lob throw-in and is immediately fouled.

Do I ignore the foul unless I determine it to be flagrant or intentional? Is the ball dead as soon as it's deemed that A1 caught the ball instead of tapping it?

No rule makes the ball dead here. Call the foul as you would any other time until the horn blows to make the ball dead.

If the foul happens after he releases a shot (assuming he caught it first), it's a dead ball contact situation (ignore or go with a T or flagrant).
The rule says they can't catch or shoot, it doesn't say they can't catch and be fouled trying to shoot.

Exactly. A1, seeing and anticipating the imminent contact, may have grabbed the ball instead of tapping it as a result. B shouldn't be able to foul to prevent A1 from tapping the ball.

The ball doesn't become dead until 0.3 seconds after A1 touches/grabs the ball.

Here's The Catch ...
 

If Team A is in the bonus, A1 shoots a one and one. If Team A is in the double bonus, A1 shoots two free throws.

Disagree. Nothing says A1 can't start the shooting motion in that amount of time.

5.2.5 SITUATION C: With three-tenths of a second or less left in a tied game, each team is in the bonus. Team A has a throw-in on the end line near their basket. A1 throws the ball to A5 on the near block who catches the ball and quickly shoots. A5 is fouled by B3 just prior to the period ending horn. The ball goes through A's basket. RULING: Since A5 may not control the ball and attempt a try with three-tenths of a second or less, A5 cannot score a goal and is not considered in the act of shooting. However, B3's foul cannot be ignored and is considered a common foul. A5 is awarded a one-and-one bonus free-throw situation with the lane cleared. A5's free throw attempts will determine if the game is over or if an overtime period is necessary. (4-41-6)

Not the way I am reading 5.2.5 C

NCAA is different. Ignore the contact unless I or F.

AR 132

If he catches it, he cannot shoot it and have the goal count. So, if he is fouled, he cannot be considered to be fouled in the act of shooting. And the OP said 'catch'.

I don't know how much clearer you can get than 5-2-5...

When play is resumed with a throw-in or free throw and threetenths (.3) of a second or less remains on the clock, a player may not gain control of the ball and try for a field goal. In this situation only a tap could score.

Piling on?

Twist And Shout ???
 

New twist. What if we add this to 5.2.5 SITUATION C: The ball doesn't go through A's basket, but instead B4 commits basket interference?

If it hadn't been for the case play, which states "A5 cannot score a goal and is not considered in the act of shooting," I would maintain my opinion. Nothing in the rule itself says he can't be in the act of shooting, only that he can't score. "May not ...try for a goal" is up to interpretation. However, the case play makes it pretty clear.

So, apparently, it can get a lot clearer than 5-2-5. In fact, it does in 5.2.5C.

Edit: This is a long winded way of saying, "I was wrong."

The ball would be dead prior to the BI, so it wouldn't affect anything.