Top of Outlet not Pressing against wall

This style center screw extends into the outlet: it's possible both parts need replacing. Pull on the cover to add tension when removing the stripped screw.

Turn off the juice before taking the cover off.

Your new 'in service' cover comes with a foam gasket: do not use caulk.

Your existing box is probably fine, but when things are open evaluate if caulking is needed around the box (not the cover, the wall box).

But this is outdoors, right? That outlet should be 'GFCI' protected. And for that I recommend you hire a professional, or find a handy neighbor/friend/relative. Typically a GFCI outlet is placed right in the box, and it uses one large square opening, not the style shown in the picture. Other configurations are possible.

Note that if you can't put in a grounded outlet, a gfci outlet is an acceptable alternative for safety. I've done that at several places in my old house. When installing without a ground, there's a sticker that should be applied essentially saying "yes, we know there's no ground, don't panic if the tester tells you so but also don't expect to use anything plugged in here as a ground."