I scratched my NBA 2k13 game disc and didn't install it into my Xbox 360. I asked my friend to download his NBA 2k13 copy to my usb flash drive. Is there a way I can install 2k13 to my Xbox 360 from the flash drive?
No. If it was possible, it would be extremely easy to pirate games using hacked ROMs. What you are attempting to do is also illegal, because you are attempting to duplicate your friend's disc. Your friend didn't "buy the game" and its binaries, but rather he bought the right to use those binaries on that specific game disc. Think of it like using a fake ID.
The only way to legally get a new copy of the game is by buying a new one.
You can do this and it is legal as long as you can put your scratched disk in and it tells you what the game is. You can have your friend install it onto your hard drive or just move it onto a flash drive. This is a copy of the disk, not disk specific which is legal to do and the reason being is you can't just go to the game and play it.
It will tell you to insert your disk which you will want to do, but the first time you play the game instead of hitting "play game" as you usually would, you will need to hit options and then hit play game. Done correctly you won't hear your game spin because it's playing from the installed copy your disk; it just confirms you own a legit copy to play the installed disk file from hard drive. If you hear it spinning just dashboard then do it again. Once you do it correctly you can just hit play game as you usually would.
Note: if your disk is unreadable or just says play game and doesn't work then buy a new one is your only option, but if it reads what the game is you're in luck.
Have your friend come over and install his copy of NBA2k13 onto your hard drive. As long as your game can still be read by the 360 you're in the clear, Also this particular method is 100% legal and is allowed by microsoft.
Source: First Hand Experience and Community Ambassadors.
EDIT: Also confirmed with Xbox Support via twitter.