A1 and B1 fighting for the ball after a long rebound, both grab the ball and A1 steps out of bounds. Award the ball to B or go to the arrow?

B's ball.

A1 caused the ball to go OOB so it is B's ball. NF 7-2-2.

Unless you deemed that a held ball occurred prior to the player stepping OOB.
It is a judgment call.

You have to go with the first action that causes a dead ball:
6-7-3 tells you that a held ball makes the ball dead
6-7-9 tells you that a violation, such as the stepping OOB, makes the ball dead

Yes whoever is the first to step out of bounds it is the other teams ball.

Give the ball to B and roll on.