Yes, if you have the patch from Microsoft that lets you play X-box games on X-box 360.
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Unfortunately, No. Here is a list of all Original Xbox games playable on the Xbox 360. http://www.xbox.com/en-US/games/backwardcompatibilitygameslist.htm
No, original XBOX controllers do not work for the XBOX 360.
No they do not, but you can play certain Xbox original games on Xbox 360
No, but some original Xbox games work for Xbox 360.
some original xbox games can't be played on an xbox 360