I'm never going to find the rules reference (could be in any of the Cryptozoic deck-building forums), but I'm pretty sure player chooses if there's a tie.
Agreed. At least in this case (not certain about every card).
While there aren't a lot of similar examples to look at for precedent, I'd say Captain Boomerang(C3 Character) is a shining example of "pick one".
"Once per turn" effects are always optional: (do these always have 'may'? not sure)...
I suspect that, for everything else, it's mandatory unless there's a 'may'.
"Once per turn" do not always say may. The HU locations are a good example (they do say 'may' but not where it counts for this question).
The ones that do NOT use 'may' are generally always helpful effects, so it'd be a corner case where you wouldn't want to do it.