Base Coach Assisting Runner

 
1. With 2 outs and the bases loaded, the batter (B1) singles to right field. The runner on 3rd base (R3), easily scores. The runner from 2nd base (R2) slows downs as he touches 3rd base, so the 3rd base coach grabs R2 and then pushes R2 toward home. R2 easily scores and then the runner from 1st base scores as the ball is wildly thrown around the field by the defense. What is the call on this play?

Call
If a coach physically assists a runner in his advance or return to a base, the coach has interfered. The runner is out, but the ball remains live until the end of playing action.

Since R3 had scored before the coach's interference, his run counts. R2 is out because of the coach's interference. This is the 3rd out of the inning, so neither R2 nor R1's run count.

2. With 2 outs and the bases loaded, B1 triples deep to the fence. As R2 and R1 round third base, the third base coach "high-fives" the runners as they pass him and then they both easily score. Any call needed on this play? 

Call
Simply touching a runner does not in and of itself constitute coach's interference. This does not appear to be interference, so R2 and R1's runs count.

Bonus Question: The 3rd base coach's son (B1) hits an over-the-fence home run. As B1 rounds 3rd base, B1 trips over 3rd base. The 3rd base coach (Dad) runs over to B1, picks him up, dusts him off, and pats him on back. B1 then runs home to complete his home-run-trot. Any call needed on this play?

Call
An over-the-fence home run, is a dead ball and a 4-base award. This type of interaction on a dead ball is not interference, so B1's run counts.