How to unscrew screws that are barely accessible

You're looking for a bradawl:

Its business end is like a straight slot screwdriver, only typically smaller and sharper. You push it into the wood where you want the screw to go, then twist to open a small hole. The screw tip will then fit into the hole without slipping. Be careful not to let the tip of the awl slip as you're pushing it into the surface, especially as you said it's slippery.

You might also get away with using a scratch awl, which has a pointed end instead, so it will make a smaller hole. If the hole is too small, you still risk having the screw slip out of it.

I've found that using a hot air paint stripper works.

Use the concentrating nozzle and a low to medium setting. Then when the paint is soft use a flat bladed screwdriver to first take the paint off the surface and then from the slot itself.

Once the head is free from paint a further heating of the screw will also help as there will be a differential expansion of the screw and hinge (or whatever the piece is) which helps loosen the screw.

A further trick I don't always remember is that it can help to try to tighten the screw first before removing it. This might just be enough to break the rust/paint/whatever is causing the screw to stick.