Hate to tell you this, but every ps3 can play ps1 games including the slim.
20gb 60gb 80gb are backwards compatible with ps2 games.
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First Generation models ie. those that were released first were fully backwards compatible and played PS2 and PS1 games as well as PS3 games. The second generation of the PS3 also which also had 4 USB ports were also Backwards compatible with many of the PS2 games. All other PS3 models lack the backwards compatible feature and are not considered to be Backwards compatible. see related link.
yes No you can not turn a PS3 into a Backwards Compatible PS2 game player. That's why the demand for older Backwards Compatible PS3 models with the 4 USB ports has increased the resale value. If you could make a Backwards Compatible PS3 Slim 250 GB why would someone pay as much for an old PS3 with an unknown history that might have been repaired many times.
NO it is not. The only PS3 80 GB model that was backward compatible was CECHE. It was the last PS3 model that was backward compatible. Even the PS3 40 GB models CECHG and CHECHH were not backward compatible. The related link for Playstation support provides details on the different models
yes
Yes Both 60GB models are, although the first is the best for PS2 game compatability