Endline Running...
 

After made basket by A1, B1 takes ball out and after 4 seconds of trying to make the inbounds pass against the press requests TO...it's granted.

We come back after the TO, they can still run the endline...

On the throw in, the ball is thrown toward B2 where it is touched by both B2 and A2 and batted back out of bounds by A2 who was the last to touch it.

B is still allowed to run the endline or no because they touched the ball on the playing floor and since A knocked it out, it's now a spot throw in?

I though I remember reading a case play awhile back where I thought it indicated running of the endlined ceases but I don't recall all the specifics....

The way I'm reading it, that's an out of bounds violation by A2, B gets a spot throw in nearest where the ball went out.

They are running the endline in your first situation.
The ball is oob off A2 in your second, B's ball
B's throw in is a spot throw in, although whomever it went off doesn't matter. Once the ball was touched legally inbounds you have a spot throw in.

B cannot run the end line. 7-2-2 - The ball is caused to go out of bounds by the last player in bounds to touch it or be touched by it (this would be your A2). Then 7-5-3 Designated out-of-bounds spot throw-in nearest to where the ball became dead (this would be where A2 knocked it out of bounds, not the end line). The result is a spot throw-in, not an end line run.

Thank you sir...

B can't run the end line.

5-7b A team retains this privilege if the scoring teams commits a violation before the throw-in ends

4-42-5 The throw-in ends when the ball touches or is legally touched by another player