If it doesn't involve a joystick on the PS2, yes,yes you can.
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Yes you can
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
Yes if you have a Ps2 which can play both or, if you have a Ps1 you can only play the Ps1 game
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, but I found that not on all games some ps2 games require dual shock 2 controllers.