1) I have a card that says "at the beginning of the villain phase, draw a card from the villain deck." A number of cards say something like that, so the specific one doesn't really matter. The card I draw says "at the beginning of the villain phase, do XYZ". Do I do XYZ on the same turn I drew the card? Are we still in the beginning of the villain phase, or did that end when I followed all the directions on the card that I already had in hand?
Is this true for all "beginning of"/"end of" directions? Now, if you draw a card at the beginning of the phase that says do something at the end, I would think you would do it on the same turn.
2) If I'm in a situation where I'm supposed to apply damage or healing to the hero with highest/lowest hit points, and I have a tie, do they both take damage or get healed? Do the players get to decide which one?
1) Start-of-turn effects played during the start-of-turn phase take effect that turn. End-of-turn effects played during the end-of-turn phase also take effect that turn. You process start- and end-of-turn effects in the order in which they entered play. Any new effect necessarily enters play after the card that caused it to enter play, so it still has a chance to be processed. The only exception (and this isn't really an exception) is start-of-turn effects that enter play after the start-of-turn phase; at this point, start-of-turn effects are no longer being processed, so the new one won't take effect until the next turn.
2) One of them will take the damage/healing. Players choose how to break the tie.
1) Yep, you're still in the Start phase, so if your new card says to do something during the Start phase, you have to do it. It's more common for this to happen during the villain's End phase, but it's the same thing.
2) The players get to choose in a tie like this, so you'd just pick which one you want to take the damage or the healing.
EDIT: dangit, I even refreshed before clicking "submit" to make sure I wasn't going to get ninjaed. Some ninjas are just that good.
We were 50/50 on our guesses on how to play. We had been doing the start of turn/end of turn correctly. We have been giving both heroes damage if they were tied, so that's a relief.