Bathroom mirror fogging up even when bathtub is not in use

This turned up a few sources --> Googling "Mirror Screw Covers"

And at least one place calls these "grommets with covers" --> CRL Mirror Grommets

Foul odors are usually due to the water seal in the P-trap failing for some reason. Sometimes the water simply evaporates in little used fixtures. Probably not the case here. The other cause for failure is the lack of proper venting. Without a proper vent, the water seal gets siphoned out of the P-trap after the fixture is used.

If you have an air admittance valve serving as a vent, it could be failing to open properly, or it could be stuck open, allowing sewer gases to enter the room via that path rather than the P-trap. These devices should be accessible. Simply manipulating the valve disc by hand can dislodge it from it's stuck position and allow it to work again. Or the device might have failed and needs replacement.

You quite possibly do not have an AAV, but the more common vent through the roof. This time of year in Northern climes, the vent could become obstructed by ice or snow. Simply clearing the blockage will solve the problem. It's also possible a bird decided the vent would make a nice warm nesting site. Other vent problems could be more endemic of the installation itself, being either undersized or even non-existent or otherwise ineffective, or an alternative, less effective, trap arrangement such as S-traps or drum traps. If you've lived there for a while and the odors just started happening, these endemic issues are probably unlikely.