Yes you can. The PS3 should have come with regular red, yellow, white adapter. If it didn't you can pick one up at any local electronics store, i.e. Best Buy, GameStop etc...
You may also want to see if your tv can do component. This is basically a high def version of the red yellow white setup. How old is the tv?
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The old PS3 can only play the oldest PS3 games. A new PS3 can play both new and old games.
No it is not free to download PS2 games unto a PS3. Most PS3 can not even play PS2 games but even the old 4 USB Port PS3s that can play the PS2 games can not download them for free. Even the old PS1 games that the PS3 can play are sold at the Playstation Store. Sometimes Playstation Plus customers are offered free games that expire with there Playstation Plus membership
Yes, usually an old PS3 disk should work on a new PS3 console.
The PS3 had some old models that could play PS2 games and if you had one you would not need to ask this question
No and only the old PS3s with 4 USB ports that are a few years old could nothing newer can play PS2 games