On-Deck Batter or Umpire Handles Live Ball

 
1a. With a runner on 2nd and no outs, the batter singles, and the throw from the outfield is late (and wild) at the plate. The runner scores, the batter is heading to second, and the catcher is chasing down the wild throw --  when the on-deck batter picks up the wild throw and hands it to the catcher. What does the umpire do at this point?
 
1b. What if the on-deck batter is hit with a throw but the on-deck batter is attempting
to avoid the ball?
 
Call
1a. By a strict interpretation of the rules, an offensive player has handled a live ball and somebody should be declared out for offensive interference. The more practical interpretation is that if runners had stopped attempting to advance, the umpire should call time and not penalize the offense. If runners were continuing to advance and the catcher had a play on the runner, time should be called and that runner should be declared out.
 
1b. If the on-deck batter is inadvertently hit by a live ball -- play on!

 
2a. With a runner on 3rd, the catcher's return throw to the pitcher tips off the top of the pitcher's glove, so the runner at 3rd bolts for home, and looks to be able to easily steal home.  However, the helpful field umpire (who was in the "C" position) fields the overthrown ball and returns it to the pitcher. What does the home plate umpire do?
 
2b. Is this handled differently if the return throw is so poorly thrown that it hits the field umpire
instead of hitting the pitcher's glove?
 
Call
2a. In NFHS, as soon as an umpire handles a live ball, the ball is dead and the runners return to the last base touched prior to the umpire handling the ball. In this case, although the runner would have scored, he is returned to 3rd. When the offensive coach comes out to argue, do not be quick to eject him -- since the umpiring crew caused this problem!
 
2b. If the umpire is hit by a live thrown ball -- play on!

 
3. With a runner on second and a 3-0 count on the batter, the pitch is wild and goes all the way to the backstop. The batter sprints to 1st base (thinking he might get two bases on the wild-pitch ball 4) and the runner at 2nd base sprints for 3rd base. Before the catcher can chase down the wild pitch, the helpful plate umpire takes a new ball out of his ball bag and hands it to the catcher. What do you do now?
 
Call
3. The umpire should only introduce a new ball into the game when the ball is dead (for example, after a foul ball or when thepitcher asks for a new ball). As in #2, the ball is dead and the runners return to the last base touched prior to the umpire handling the ball.