Ruger American Rifle with Vortex Crossfire II Riflescope Model 16934 in .308. Forget about a wooden stock. MSRP is $639 but I've seen them on sale for less. Even at MSRP you get a very good scope, already mounted, so don't stress on your budget. This is a very good rifle for the money.
First Gun
First of all: newcomers tend to think that pistols or revolvers are the easiest to start out with. That's wrong. Pistols and revolvers are the hardest firearms to shoot with, because they have a much shorter barrel, and you can't correct your aim with one hand further forwards. Get a rifle with a barrel that is at least as long as your forearm, preferably longer.
Rifles also tend to be a heavier than pistols or revolvers, which makes recoil a lot easier to handle.
Depends on what you want to hunt. Small game, large game, clay pigeons, or real pigeons?
Any sort of rifle. As I said, pistols are much more dangerous for newbs. Check places like gunbroker or your local friendly gunshop.
So a .22LR or 5.56mm rifle. That's always a good choice. Many consider them Babies First Gun, and for a good reason. They are easy to maintain, easy to handle, easy to acquire and don't require a large investment for ammo or the gun itself.
Depends on how active you want to be with the gun. If you want to get into something a little more hardcore you should probably consider polymer stocks because they are a lot more durable. But wood is good, especially for a beginner. I also prefer the wood A E S T H E T I C S.
Yeah, a .22LR is well within your range.
NONONONONONO! Fuck that. You don't just lock your gun in your car. All it takes is for someone to see you put it in, wait until you are gone, and smash your windows. Congratulations, you put one more firearm into the hands of a criminal.
NEVER EVER be more than 20 meters away from your gun without it being secured to some solid surface or in some sort of safe or case. If you are not using it, keep it locked and unloaded. If you really don't want anyone to know that you have a gun in your dorm, get a takedown rifle (rifle that can be easily disassembled and stowed) and keep it in a locker underneath your bed. When you take it to the range put it in your backpack before you leave the room, and don't take it out until you are at the range/innawoods.
Alternatively: don't get a gun until you have your own place and make your own rules.
Before you buy I want you to look up the following things:
What is a caliber, and which one does your gun use?
What are the four rules of gun safety, and why should you know them by heart?
What are the regulations about carrying a firearm in your state?
What is a QTDDTOTT?
Why should you consider to post questions such as this in a QTDDTOTT in the future?