That depends entirely on state and local laws. There is no answer good throughout the country.
Nothing. There is no gun registration in Virginia.
Pennsylvania doesn't require licencing to purchase ordinary Title 1 firearms. For a concealed carry permit, you must be 21.
If the ownership of the gun is changing, it must be sent to a firearms dealer in SC. If the ownership is not changing (owner is moving to SC) no action is needed. Neither state requires registration of guns.
Anne Hold That Gun
yes
Guns are not registered in the state of Virginia. See http://www.vsp.state.va.us/Firearms.shtm for more information.
Acrylic and silkscreen on canvas. The title is simply 'Gun'.
Gun Control
In short: no. As a matter of fact, New Jersey is one of the most strict states in the US in terms of gun control. Off hand, only Washington DC has tighter gun control (not allowed to even OWN a gun in your own home). You'd be well advised to not carry a weapon into New Jersey. If caught, well, let's just say that they don't take too kindly to guns there.
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
Pennsylvania doesn't require gun require registration, and even in states that do, the average person can't access the information.