Actually that's incorrect. If a firearm is stored in a safe you can store ammunition and loaded magazines next to it, same goes for storage in cabinets.
Separate storage of ammunition only applies if you keep a rifle or shotgun trigger/cable locked and not in a locking container. So long as they're disabled with said locks you can just store them under your bed, but ammo needs to be in another room or in a locked container.
Ahh, that would allow one to more quickly respond, but still requires them to keep their safe in their room and still prevents them from being able to defend theirself when not at home (or should someone break in when they aren't close to the safe). And to me that would still be pretty bad as it still greatly restricts one's ability to defend theirself.
I heard you can get a carry permit to carry on base? I wont be reserves, so I'll most likely live on base. Unless, can you get an apartment off base if you're single? (I know that you get free lodging on base but I'm not sure if living off base is an option)
You can't get a carry permit to carry on base. Even security forces and OSI can't carry while on base unless they are on duty and the gun was issued to them from the armory (at which point they have to turn in the gun once their shift ends).
As for storage, it depends on your service. I've heard of Army/Marine barracks having their own mini-armory at the actual barracks for weapons (dunno if all of them have this though), while in the AF (and I would assume the Navy as well) you have to use the base armory. Firearms are not allowed to be stored in the barracks or at the base hotel (I had the cops called on me for that one when I simply asked the hotel manager their policy, but that's another story...). If you live in actual base housing then you can store the weapons at your house, but there's requirements for doing so. In addition, any weapon stored on base (whether at the armory or in on-base housing) requires paperwork. I can't speak for other services (though I assume it's very similar if not the same), but in the AF the paperwork has various info about the weapons and must be signed by your commander specially stating where the weapons are being stored.
For your final question. Each base and service is different in regards to when one can move out of the dorms. You would need to get to your base and ask the question to your leadership, but in the AF you generally have to be at least a Senior Airman (E-4) before being able to move off base when single (or married, but don't get married to move off base as people do get kicked out for doing that). Oh and once you get to the point to be able to move out of the dorms know that if you decide you want to try and live in base housing there's generally a waiting list (but again this is base dependent and the length of the list varies from each base).