I think my GTA disc is scratched and the game often restarts when I'm doing missions. It only happens during missions though. But i was thinking if I installed the play disc it wouldn't be spinning anymore and so the game wouldn't restart? However im not sure whether to do this, will It fix the problem and if if it doesn't can i uninstall it without losing my GTA Online progress??
You can install Disc 2 and it will likely solve your problem, assuming the problem really is because of scratches on the disk.
The reason why Rockstar does not recommend installing both, is to boost loading speed; after all, loading half the game from HDD and the other half from disk is faster than loading everything from HDD.
In the case that installing disk 2 does not solve your problem, you can easily uninstall the game without losing any progress, since game data and save games are stored separately. In the case of GTA Online, your character is saved on Rockstar's own servers; however, it seems like Rockstar servers don't like to remember people, but this is completely unrelated to installing/uninstalling the game on your console.