hmm... looks like it was an option on the xb360. not sure why you'd want to do that though.
are you referring to the 4gb internal storage the xb360 came with on newer models (its internal memory) or the expansion drive they sell for it?
internal memory is required absolutely as it is not just used by games but also for system updates, dashboard, etc, but the removable upgrade is completely optional, yes. this is why many of the newer models were sold without hard drives. its optional for them.
if the game requires no install to work then yes you could make do without one (though loading from the disk would be horrible and frequent load times). gamesaves would be saved to your internal memory. generally its best to have the game...