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The Xbox original does a hard drive like any other console. The Xbox 360 however is portable and can be changed.
On the older Xbox 360's, at the top. The Xbox 360 S, the hard drive is located at the bottom of the Xbox.
Yes, if it is detachable, and it will have all of your game information on.
The Xbox Arcade is an Xbox 360 and it comes without a hard drive. What you are calling an Xbox 360 I'm assuming comes with a hard drive and if you are thinking of buying an Xbox I recommend not buying the arcade as you cannot save games without a hard drive.
In short: no. The Xbox 360 has to install an original Xbox emulator program to run the Xbox games. It stores this emulation software on the hard drive.