@ulrich: thanks! I forgot that blowing up makes them disjoint. Another edit...

@Dustin Cartwright: Thank you! I tried to use this argument for the general case, but then it's difficult to control what happens at the boundary after you take the closure - I didn't notice that it works in the curve case. I edited the question.

Christian, I don't think this can work - by a result of Elkies, an elliptic curve in characteristic 0 becomes supersingular at infinitely many primes, so this would contradict (2).

Dear Jörg, I cannot find your script on your homepage. Are you going to post it there? I would really like to see it!

An obvious idea is take $Y$ to be a blowup of the fixed point set (or the set where $G$ does not act freely). I think this works for Kummer surfaces: if $A$ is an abelian surface then its Kummer surface $X$ is a crepant resolution of $A/i$ ($i$ is the involution $x\mapsto -x$ on $A$), but can also be described as $B/i$ where $B$ is the blowup of $A$ along the $2$-torsion subgroup (i.e. fixed points of $i$).

In case when the series is a solution to some linear differential equation, this seems to be related to en.wikipedia.org/wiki/… .