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Will they still make games for the old ones, or will the XBOX, PS2, and Gamecube be nothing more than expensive paper weights

I say it will be somthing in between. There are still a few lengthy projects in the works for the current systems.

Judging from history, it'll be a while before they stop making games completely - I think the last PS1 game was made in 2002, in fact

Odds are next gen consoles will be backwards compatible, it will probably become a trend from now on and so yes it will make older console development much longer run thanks to being able to play on new consoles.