If you mean a legal resident alien, yes. If you mean a legal resident of Arizona, yes. If you mean a legal resident of another state- THAT depends on which gun. You may purchase a rifle or shotgun from a dealer in any state (unless you reside in CA) but you may only purchase a handgun in your home state.
Yes.
how to become a arizona resident
Yes the background check for guns is the same in every state
No, you can divorce in Arizona if you are a resident.
Yes... I may be wrong about this, but I believe Delaware is one of the states where you must be a resident of that state to buy a firearm from there, although I could be wrong about that.
Yes- but only from a dealer. You may not buy a firearm from a private seller in ANY state other than your own.
If you are a resident of Virginia, and not prohibited from owning a gun, you may buy a gun from any other resident of Virginia. This applies in MOST states. If you are not a resident of Virginia, then you may buy a rifle or shotgun from a Virginia dealer, but not a private citizen. This is addressed in Federal law- US Code 18.922
A resident of one state may buy a rifle or shotgun from a licensed dealer in another state IF his home state approves such a purchase (Kentucky will). You may not buy from a private citizen, dealer only, and you cannot buy a handgun in another state.
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Arizona
You only have to live in Arizona for 6 weeks to be considered a resident. If you have a home and receive mail there, you are considered a resident.