Where to spot the ball
 

Team A drives to the basket and team B steals the ball takes one dribble and throws the ball out of bounds. The ball crosses over out of bounds at team B goal should the ball be placed at the baseline or where the player last released the ball. The player was in backcourt when he released the ball if that maters.

Ask yourself: When does the violation occur? Where is the ball located when the violation occurs?

Answer these questions, and you will have your answer.

Violation occurs when ball goes out of bounds. Inbounds takes place there.

Correct.

Minor quibble.....
Ball is put in play at the spot where the ball first touches OOB

Where it touches out of bounds is where it goes out of bounds. The throw-in spot is the spot nearest to this.

We both know that but the OP seemed to be improperly fixated on where the ball crossed the boundary.

What if it hits the ceiling? ;)

I only asked the dumb question as the game I watched the ball got placed at half court with 10 seconds in the game with team A leading by one as that is where the player last touched the ball. Team A coach was upset as half the people in the stands.

It happens. Refs miss calls, or misapply rules, or out think themselves. Getting upset about it is pointless.

If the ball is being put into play from the baseline and does not touch a player inbounds before going out OOB at the opposite baseline, it returns to the original throw in location, correct?

That is correct. The throw-in ends when it is legally touched by another player. Because the ball was never touched and went out of bounds, it is given to the other team at the original throw-in location.

If we remember what the violation is, it becomes easy(ier) to remember where to put the ball in play.

In this instance, the violation is not for the ball going OOB, it's for "failing to pass the ball so it's touched by a player" -- so the inbounds spot is the spot of the original throw-in