Where is the proper place for a throw in to occur after a back court violation: at half court or the spot where the ball was actually touched by the violating team?

The violation is for "first touching" the ball. Thus, the throw-in is at the spot nearest the touch.

There wasn't a violation until the ball was actually touched in the backcourt. Thus, the designated spot should be at the spot OOB nearest the violating touch. (Case 7.4(a))