the larger ps2 you can ...yes not the slim ps2
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yes, the ps1 controllers are dualshock, and the ps2 are dualshock2, so it may not be as good overall as a ps2 controller, but it should work
Can't play ps2 games or use ps2 memory cards.
Of course you could, the jack on both Sony PS1 and Sony PS2 are exactly the same, although it lowers the performance than if you were playing with a Sony PS2 controller which means that you can not actually use the controller because who wants a controller that does not work completely. Used PS2 controllers are cheap and there are no good reasons to try and use a PS1 controller on your PS2 because it will not preform like a PS2 controller
Yes, only for PSone games though. PS2 games require a 8MB (PS2) memory card.
No, PS2 memory cards are keyed differently, and will not physically fit inside the PS machine.