I wanted to know if it's possible for me to play a game disc I have by installing a backup of the digital version on a usb, even if it's damaged
From what I remember this doesn't work with the marketplace version of a game on the Xbox 360. It does on the Xbox One / Series consoles however. On the 360 the game needs to be installed from DVD, so you can borrow a friends disc and install the game. After that you can then used your scratched disc to play the game, presuming the 360 can read it enough to recognise what game it is.