I am trying to repair my old analog camera. I have mirror that occasionally locks up, which can apparently be fixed with a drop of oil. For this I have to remove the bottom plate. Apart from a few simple screws, it is held back by the rewind button (shown in picture). It seems like it's easily removed by jamming two pins in the holes and twisting left, but I'm not gonna risk forcing it in case I'm wrong. What's the proper way to remove this?
You could use a lens spanner wrench. It may be difficult to find one that accommodates such a small diameter, though. Personally I'd try the two pins method you suggest first – not much force should be needed, so probability of damage is low if you're careful. Maybe you could fashion a tool out of a paperclip? Another trick that sometimes works with similar buttons is to apply friction as you turn, using for example a piece of rubber between the button and your thumb.
(Are you sure though that the bottom plate is held back by this button and not by the battery cover?)
Yes, that is removed by a caliper. It will disfigure very easily so don't guess at it. If memory serves it unscrews clockwise, but you haven't given the camera model so don't hold me to that. BTW, you shouldn't 'oil' a sticking mirror. You will just add to the problem. Clean the lever's movements with lighter fluid or electronics cleaner and cotton swabs but don't overdo it. Excess fluid will run onto the shutter. It's not a lack of oil that makes it stick. It's a build up of old grease and gunk that causes the problem.