Does the 10 second count start when the ball is touched by a player after a throw-in or with player control after a throw-in? Does the fact that in NCAA rules team control exists during the throw-in affect this rule at all as compared to Fed?

In FED 10 second count starts when there is team control int he backcourt.

In NCAA, team control exists during the throw-in, but the 10 second count doesn't begin until the throw-in is controlled inbounds.

as an aside, in games that involve a shot clock, the shot clock begins when the ball touches any player inbounds

Which is why you can't always use the shot clock to determine your 10-second violation. The shot clock starts on the touch, but the ball could be batted for several seconds before it's controlled. So your 10-second count might not even start until the shot clock was down to 27.

(NBA's a different story, of course. Since there's team control during the throw-in, the 8-second count starts as soon as the ball is touched inbounds.)