I took the rear wheel off and it appears to be a freewheel that takes the 4-prong SunTour tool.
This isn't mine, but it my freewheel looks essentially like this -- just a lot dirtier, lol. What would the difference be? I figured that in either case I'd be buying new gears, a hub, and shifters, just that the gears and hub would be a single piece in the freewheel scenario. Clearly my knowledge is incomplete, though.

Also, I am assuming in all these scenarios that my derailleurs will work fine with new components. My understanding is that the derailleurs simply move however the shifter cable tells them, and that as long as they are capable of sufficient range of motion that they can be tuned with cable adjustments and whatnot. Once again, however, I am assuming that my knowledge is incomplete.

BTW, thanks so much for all the great information, everyone! I am definitely going to attempt to clean (and maybe even take apart depending on how the cleaning goes) my old freewheel before I replace it, and if I do replace it I have a much better idea what my options are.
Such a helpful bunch