It depends, but usually yes. The problem is that sometimes usb drives are used for piracy to simulate having the CD. If a game checks if it's running from a USB device and the answer is yes, you may have a problem. Otherwise, you will be fine.
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If the drive on the PS2 is broken just buy a used PS2, they are very cheap and many come with games and accessories
Yes, you need a hard drive to play original Xbox games because the hard drive contains the emulator that allows the original Xbox games to play on 360
No, games need to be installed locally onto the hard drive of the computer you wish to play it from.
In order to play games for the original Xbox on the Xbox 360 you will need to have the hard drive. This is because the update is downloaded onto it.
No you can not download and play PS2 games that way.