[BktBallRef]

"A guy is is claiming that you have to both signal a timeout with your hands and call timeout before it will be granted."

He's full of crap. Tell him to cite the correct rule reference.

NFH 5-8-3
Time-out occurs and the clock, if running, shall be stopped when an official:
Grants a player's/head coach's oral or visual request for a time-out, such request being granted only when:
a. The ball is in control or at the disposal of a player of his/her team.
b. The ball is dead, unless replacement of a disqualified, or injured player(s), or a player directed to leave the game is pending, and a substitute(s) is available and required.

"B1 is falling out of bounds why does he have possession enough to call a timeout but not possession enough for the shot clock to reset if he saves the ball to team A or falls OOB?"

If the defense gains possession of the ball while still inbounds, the shot clock would reset and he can request TO. If he doesn't gain possession until he lands OOB, there is no shot clock reset because team control was never established by the defense. His premise is false.