I'm aware that the 20-gig model of the PS3 has no wi-fi capabilities; however, the Ethernet connection I have is virtually nonexistent. If I use a wireless USB adapter to plug into my PS3, will it give me Internet functions wirelessly?

I've got a 20 gig as well, what do you mean that the Ethernet connection is virtually nonexistent? Mine works great, haven't tried the wireless adapter method though.

It constantly says the port is not connected when I have the cable hooked in.

Don't know about your ethernet issues, but the main weakness of the 20G is that the PSP won't connect to the PS3 for remote play using 3rd party wireless.