When the Xbox says install game it does not mean you are about to install the full game onto your hard drive. When you install a game on to your Xbox it's taking some of the game files and installing them to make the game run smoother and possibly look better.
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It basically cuts the loading time on the game etc...
tooth paste? are you kidding? you cant fix a game with tooth paste
Your disc may be scratched, you can fix that by borrowing the game from a friend and install the game (from the game library tab) and then return it and it should work.
It's quieter AND uses a crap load of space.
Probably the game is old, scratched, dirty, etc. Also if the game system is old, scratched, dirty, etc. Or if the power source is old, scratched, dirty, etc. Also there can be something that went wrong in the creation of the game. Also, if you are playing an online game, it could lag because you have bad connection to the host.