Q1:
Runner on 3rd with 1 out. Batter' swings and his bat ticks the catcher's mitt while hitting an infield grounder. The shortstop's throw beats the batter at first. The runner from 3rd crosses the plate safely. What's the ruling?
a) Catcher's interference. Batter is entitled to 1st. Runner returns to 3rd.
b) Batter-runner is out, run scores.
c) Offensive team's manager chooses between a) and b).

d) Offsetting penalties, start over from the beginning.

Important: What's your thinking (in your own words, not rulebook quotations) for your choice of the correct ruling?

A1:
The answer (by one of the winners) is C.
In the first scenario, the catcher has committed interference against the batter. It is the offensive coach's prerogative on whether to accept the penalty against the defensive team or decline the penalty.
If he accepts the penalty, the ball is dead at the time of the interference, and the runner on third returns to third base, and the runner is awarded first base.
If he declines the penalty, the batter runner is out and the runner on third scores.
The winners are: Rick Oherron, Cliff Baker, Toshi Takeuchi (Manager/Coach); Michael J Rizzo, Jeff Hobson (New Umpire); James Wheaton, Oliver Konkel, Ronald Ward, David Paul, Alan Miller (Veteran Umpire); and [honorable Mention--gave a good answer but was not directly based on the applicable rule] Carolyn Watson (Family/Friend). There were few player entries and no winners.
Everyone: Please be reassured--ALL the veteran Umpires answered correctly. The majority of the managers did too.

 Q2:
Two outs, runner on first, game tied and in extra innings. Batter for home team hits a clean home run over the centerfield fence. The runner from first (R1), bouncily exulting in the belief that his team has won the game, rounds second but does not touch the bag. He realizes this after he passes the bag and scampers back; the batter-runner (BR) arrives and touches the bag before R1 gets there. R1, never actually touching second base, turns and heads toward home as does BR. Both touch third and the plate in proper order. Cheers erupt; perhaps--if only in spirit--a bucket of icy Gatorade is poured over the coaches' head.
Depending on several possible scenarios of what happens next, what do you A) as base umpire, B) as plate umpire do? In your own words (not rulebook quotations) how do you explain the correct ruling(s) to the managers?
Bonus Teaser (no award, but try it if you'd like the additional challenge): Same setting and events as above except that, at the start of the play, there is only one out. Any difference? Why or why not?

A2:
Using responses cobbled together from several of the winners:
The play hinges on the fact that by rule, a run is only actually scored when a player successfully touches each base in order without being put out--even on a home run.
Both R1 and the BR make mistakes here, because R1 missed second and the BR did not hold up to let him return to touch it. A missed base IS an out but ONLY on an appeal.
A player on defense has to inform the umpires they are going to appeal--neither umpire can say or do anything that signals either team that a base was missed, that it is an appeal play, or how they might rule on it. If all members of defensive team do not completely leave the field (go into foul territory), the pitcher asks plate ump for a ball to be put in play, and when (only when, because an appeal can be made only on a live ball, and on a home run, the ball is dead)) the plate umpire has put the ball in play again, the pitcher tosses it to a player covering 2nd who steps on the bag.
Only if that all happens the base ump calls the runner out. Since that constitutes the 3rd out of the inning made during the course of the play, no runs score. This game will continue into the next inning, perhaps slightly delayed to let the poor coach towel off the Gatorade. (In the bonus question, it would only be the second out, the BR's run would legitimately be counted, the game would be over.)
(Several people commented that this ruling would take away a legitimate victorious walkoff home run on a technicality and said the umpires should pretend not to have seen the missed base. Nope. The person presented with this dilemma is the defensive manager, who could easily choose to tell his team not to appeal for this reason. Umpires must apply the rules of the game correctly and evenhandedly.)
Winners for Question Two are Scott Adams, Dick Cadenasso, Jonathon Stewart, Cliff Baker (Manager/Coach--and they all got the bonus as well); Michael J Rizzo, Jeff Hobson (New Umpire, and Jeff tried and got the bonus); James Wheaton (who also tried and got the bonus correct, Oliver Konkel, Ronald Ward, David Paul, Alan Miller (Veteran Umpire). No Players or Family/Friends correctly answered this complex, multifaceted question.
More reassurance (perhaps not really needed, but)--ALL the Veteran Umpires answered this one correctly, as well as they did Question One.