I like Scully in her glasses--I think she looks great in them.
R.e. the baseball cap--I wondered this, too, but finally decided she kept it in her hand--her left hand, I think. You can see it come up at one point when Mulder has Scully's mouth between his hands. Also when she hugs him, it's certainly possible it's in that hand, as we only see the other hand clearly. If you care to, check and see if you think I'm correct. Happy re-viewing!