During a jump ball to start the game, A1 slaps the ball out of bounds. Would B get the ball and the next AP arrow? Would the arrow be set or switched when the ball is at B's disposal or when the throw in ends?

Shont

B gets the ball for a spot throw-in. Possession Arrow goes to A. B is the first team in this situation to actually have team control.

And the arrow is set when the ball is at B's disposal -- not immediately upon the violation by A and not after the throw-in ends.

Ok, I think I now understand case book 4.3.3. A in this sit. would have had the pos arrow, but B2 commited a tech foul. Therefore A will have a throw in at the division line and the arrow will then be pointed to B when the ball is at the disposal of A. Is this correct?

That's correct. Since the T occurred before the ball was at B's disposal, the arrow hadn't been set. So, A gets the first posession (on the throw-in following the T) and B gets the arrow.

If you change the play a bit "During the jump, A1 slaps the ball OOB. After B1 is handed the ball for the throw in, B2 gets a T". The ruling would be, "The arow is set to A when B1 is handed the ball. A gets two throws for the T and is awarded the ball for the throw in. The arro remains pointing toward A's basket."

What you write is only true for NCAA play.
Under NFHS rules, Team B may not actually be the first to have team control, but Team A could still receive the arrow first. Consider this: even if nothing goofy happens following the OOB violation, and you place the ball at the disposal of B1 for a throw-in, there is still no team control at this point. Now what if thrower B1's inbound pass is intercepted by A1 or if B1 violates during the throw-in? Or perhaps even a foul occurs before the throw-in ends! So you just can't say yet which team will be the first to have team control. Of course, team control is rather meaningless here, since placing the ball at the disposal of the thrower is what determines the initial setting of the possession arrow in this particular play.