No they do not. Quick reason: PS3 uses Blu-ray discs to store the games while the PS2 uses DVDs.
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NO ... but PS2 games work on PS3 consoles ...
Yes it does work
You can not play PlayStation 3 games on a PlayStation 1, but you can play PlayStation 1 games on a PlayStation 3. All of the PlayStation consoles are backwards compatible.
You cannot play Playstation 4 games on a Playstation 4 console. You are also unable to play Playstation 3 games on the Playstation 4.
No as the PlayStation 3 Blu-Ray discs are too large to fit in the DVD slot on the PlayStation 2, although you can play most PlayStation 2 games on American PlayStation 3's and around 50% of PlayStation 2 games on European PlayStation 3's.