I would use a multimeter to see what kind of voltage you have at the GFCI & at the lights that don't work. It's possible a short is diverting enough power that there isn't enough voltage left on the circuit to turn the lights on.

This sounds serious enough for professional help if you can't get it tracked down in short order.

Can you expand on the 'replacing the vanity' part?

What tool did you use to cut a hole in the ceiling? Any chance you shook a wire loose or cut through one?

It's easy enough to fold/cut the corner of the liner wrong, or poke it during install if you aren't careful. Cut a hole in the ceiling to keep the leak exposed, and see how long the temp solutions actually keep it dry.

If that's actually an open corner (no waterproofing/pan liner behind it & up the sides behind the tile 6-12"+) caulking is not going to keep it from leaking.

Demo & redo enough of the bottom section (and curb) to make sure there are no future issues.

Pictures?

Brand/model?

Check the air gap by the sink (it gets clogged)

Increase the size and length of the hinge screws in the holes that are egged out (I would suggest all). Likely there’s plenty of material nearby. Adding some wood glue super glue or two-part epoxy during replacement wouldn’t hurt anything either, and may help with future abuse.

If you need to enlarge the hinge holes, I suggest a zero flute countersink but.