no
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The PS2 Slim Plays all the PS2 games the Fat PS2 would.
At the moment, no. Sony has unfortunately cut PS2 backwards compatibility, and the "Other OS" capability from the PS3 Slim.
No the new PS3 slim can not play PS2 games. To make the PS3 profitable and have it quit selling for a loss Sony stopped making PS3s that could play PS2 games long before the release of the PS3 slim. The only reason you could ever play PS2 games on the first PS3s was the game catalog for the PS3 was next to nothing. Once the catalog of games increased they dropped the ability to play PS2 games.
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.
The PS2 slim does not have a harddrive