yes, but I found that not on all games some ps2 games require dual shock 2 controllers.
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Yes if you have a Ps2 which can play both or, if you have a Ps1 you can only play the Ps1 game
With a emulator for the PS1
In order to play PS1 games on a PS2 you need to have a PS1 memory card in your PS2, so you can save your progress. Otherwise they don't work.
Yes you can
If it doesn't involve a joystick on the PS2, yes,yes you can.