Can you play an Xbox 360 game on a normal Xbox

No, you cannot, but if you have an XBOX 360, or an XBOX 360 elite, you can play XBOX games, with an XBOX 360 Arcade, you need an XBOX 360 hard drive to play XBOX games.
No you can not play Xbox 360 games on the original Xbox. Some original Xbox games can be played on the Xbox360 but it just depends if they are backwards compatible or not.
Very few games for original xbox will work on 360. A total of 200 are currently compatible.