Throw from F2 to F1 hits batter
 

Situation;
R1 on 1B. The throw back to the pitcher hits the batter/bat who is legal. Local HS coach told me this happened to his team twice this season. One time the ump killed the play. The other time it was ruled live ball and R1 took 2B. I told the coach the ball was live. However I wasn't able find caseplay or the rule to cite for him. Was I correct? Is Fed different from OBR?

It's not covered in FED -- I'd treat it the same as OBR:

1) I f the batter is in the box, the ball remains live
2) If the batter is out of the box, the ball is dead, runners return
3) If the batter intentionally interferes in either 1 or 2, the batter is out

Mostly agree. However, if the bat accidentally contacts the ball (rather than vice versa), I'd kill it in the interest of fair play.

I understand your use of Rule 11 (common sense) to resolve this situation. Can you find support for your stance in the other 10 rules (NFHS/NCAA) or 9 rules (OBR)? If the situation is really confusing, and the batter tries to run after the bat accidentally contacts the ball on the throw back, I'll make the safe signal and say "That's nothing", so that the batter (and everyone else) understand that the contact between bat and ball is NOT a batted ball.

6.10 BATTER INTERFERES WITH CATCHER'S THROW BACK TO PITCHER
If the batter interferes with the catcher's throw back to the pitcher by stepping out of the batter's
box while at bat (no runners attempting to advance), it shall not be considered interference under
Official Baseball Rule 6.06(c). In such cases, the umpire shall call "Time" only (no interference).
The ball is dead and no runner shall advance on the play.
This interpretation does not, of course, give the batter license to interfere intentionally with the
catcher's throw back to the pitcher, and in such cases the batter shall be called out. If the batter
becomes a runner on ball four and the catcher's throw strikes him or his bat, the ball remains
alive and in play (provided no intentional interference by the batter-runner).
If the batter interferes with the catcher's throw to retire a runner by stepping out of the batter's
box, interference shall be called on the batter under Official Baseball Rule 6.06(c). (See Section
6.8.)
However, if the batter is standing in the batter's box and he or his bat is struck by the catcher's
throw back to the pitcher (or throw in attempting to retire a runner) and, in the umpire's
judgment, there is no intent on the part of the batter to interfere with the throw, the ball is alive
and in play.

OK. I'd say the same thing, NFHS, NCAA, or OBR. The umpire who killed the play may have been acting on common sense, but the rules were not strong with him, unless the batter was out of the box.

Personally, I'd keep play alive unless there was a good reason not to.