Where do patches save to on the 360?
Patches for 360 are basically saved in a cache folder of sorts on the HDD. You can delete it along with other cached data, and then you just redownload it.
Installing a game makes it run better. This is simply a fact.
Not necessarily. Halo 3 played worse when installed to the HDD; google it.
Anyway, I'm not too worried. For one thing, I have a Slim 360, and it's disc drive is much better than the older models (HDD seems to be better too).