Mudisfun

The base coach is a member of the offensive team who is authorized to be on the field, at that point in time. A base coach who is attempting to get out of the way, to not interfere with the defensive teams ability to field a batted ball, and who does not in fact interfere can go where ever he need to go to be out of the way. By moving into fair territory, he most likely is in a better position to 'interfere' with his own runner. I am not going to pick this one, unless of course the base coaches actions do in fact interfere. 
 
Now a coach coming out of the dug out? Whole different animal...

Fair ball.
The location of the hand in relation to the box is not a consideration, only the batters feet when striking a ball would be considered for an illegally struck ball. I don't know many umpires who would be watching a batters feet when he is bailing out of the box on an inside pitch which then out of luck strikes the bat versus the batter. Talk about grabbing the stinky end of the stick if you called the batter out on that one.