During a timeout,A1 disrepectfully addresses the official on the way to the bench. Just before the expiration of the timeout,the coach of Team B also engages in unsportsmanlike conduct towards the official. Technical fouls are charged to A1 and Team B coach. Are free throws awarded?

yes

Yes. This is an example of a FALSE double technical foul. There was a definite gap of time between the two Ts and the opponents did not commit the unsporting actions against each other.

You penalize the fouls in the order in which they occurred.

Team B will shoot 2 FTs, then Team A will shoot 2 FTs and be awarded the ball OOB at the division line opposite the table.

not in FIBA, each time got 2 FT's ecuals no FT's and a Jump ball.
Is this NCAA rules?

In high school (NFHS) there would be a double technical foul if the unsportsmanlike comments were addressed to each other, and were at about the same time. then the shots "cancel" each other and there aren't any.

A false double foul, or false double technical foul, is when the comments don't fit the discription above, and then the shots are taken.

Why does Team A get the ball?

You penalize the fouls in the order that they occur, as Nevada said. The last foul was the "T" on the team B coach. Possession to team A for a throw-in is part of the penalty for that "T" against the team B coach. NFHS rules.