With regards to distribution, does that ONLY affect how many times the player will look to shoot or does it also affect the number of touches the player gets in the offense?  For example, does it help a poor ball-handler against a press to have his distribution cut so he won't have the ball in his hands as much?

Turnover s have a positive correlation in regards to shots taken.

Distro is about how many plays are designed for that particular player to be the shooter. From everything I've seen and heard, "touches" aren't factored into this.

Yes, the more distro a guy has, the more he will get the ball in his hands (you can't shoot if you don't have the ball) so it is more likely he will turn it over than if he doesn't have as much distro.