No there is not
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No, Play Station is not associated with Nintendo
Currently, there is no Super Mario for PS2 because it is not a Nintendo console, unlike Wii. There probably won't be a Mario game for any other consoles either in the near future, but then, you never know.
Sort of... You can go to somewhere like GameStop and trade in your PS2 games for money, and then use that money to buy wii games. I don't know of anywhere where you can just swap games, though.
yes sonic games the closest to Mario since they used to be rivals
The PS2 Slim Plays all the PS2 games the Fat PS2 would.