Alright, here you go, broken down
---Question 1: Initially, I was thinking that Stone's +2 bonus would apply during the Shootout Plays, but I was wrong, wasn't I? My shooter can't get the +2 bullet bonus yet because I don't choose the shooter until after plays are over. Is that still the case even if there's only one dude in my posse (because I haven't "officially" chosen him as the shooter, even though he's the only possible choice)?
Correct, you have not declared a shooter yet until plays are over, so even if there's only one dude, that bonus is not granted yet.
---Question 2: So how long does the shooter retain that bonus? Or to put it another way, how long is that dude considered "the shooter"? Does the dude cease being the shooter once the draw is resolved, or does he remain the shooter (and thus keep the bullets) until a new shooter is chosen in the next round?
Per page 21 of the rulebook, the shooter retains designation until the end of the current round, not until the next round.
---Question 3: What happens if the chosen opposing dude is aced during the first round of Shootout Plays (by Point Blank or Legendary Holster, e.g.)? Does Stone's ability "fizzle" since I haven't chosen a shooter yet? On the other hand, the ability doesn't say "is aced as a casualty", so I wouldn't think Stone would care how the dude was aced...
The ability remains in play. As you are no longer opposing the dude, your shooter does not get the +2 bullet bonus, but once your shooter is declared, your shooter will receive the permanent +1 bullet bonus and permanent control point.