Well, if you want to be able to play all Xbox games (including Xbox 360 games), you will need to get any generation of Xbox 360, but if you only care about the original Xbox games, then you can get just the regular Xbox, which is about $40 nowadays. I'm not sure if there are different generations of the original one, though...
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The Xbox 360 can play original Xbox games but the Xbox doesnt play Xbox 360 games.
Yes xbox elite's can play xbox 360 games
Yes, It Is Possible To Play Old Xbox Games On An Xbox 360, However, You Cannot Play Xbox 360 Games On An Original Xbox.
You can not play xbox 360 games on an original xbox, however you can play some original xbox games on an xbox 360 console.
No the PS2 does not play Xbox 360 games only the Xbox 360 plays the games for it.