Bob wrote:

The pushbutton, all plastic, piece that opens my microwave door broke,
but I was able to get a new one from a parts place on the internet.

But, the screws that hold the metal microwave cover in place are
apparently some kind of a safety-screw, which I'm having a lot of
trouble with.

Those are called "security" screws, and if there's no local Harbor
Freight to get the set of security bits that Stormin' Mormon
mentioned, you can take a tiny screwdriver and hammer it sideways into
the center nub to remove it.