Replace GFCI in bathroom
A white wire connected to a hot can be seen on switched hots. A good electrician will often add a piece of black tape to indicate this, but that often doesn't happen.

When connecting wires to screws on outlets, you only want one wire per screw. If you need to attach multiple wires and don't have enough locations to screw them on, then pig tail them together with a short piece of wire, secure and cover with a wire nut, and connect the short piece of wire to the outlet.

To resolve the GFCI not resetting, disconnect everything except the line connections. Verify the GFCI is working properly. Then connect each load one at a time until you locate the fault. You'll need to trace these lines or identify what isn't working when disconnected and inspect every junction to locate the problem with that line. If you don't feel comfortable with this, then I recommend hiring a professional electrician. As always with electrical, do not work on live wires and use a voltage tester, preferably non-contact, to check all wiring before you touch it.