Yes the background check for guns is the same in every state
Yes- but only from a dealer. You may not buy a firearm from a private seller in ANY state other than your own.
No
Rifle or shotgun FROM A DEALER- yes. You may not purchase a handgun except in your state of residence, and any long gun purchased in another state must be from a licensed dealer (not a private person). The answer to this question has changed over the past several years, at one time you could not.
yes
No.
No. You can purchase a gun from a Georgia dealer, have it shipped to an FFL licenced dealer in California for receiving, and receive it in California, but you can't purchase a gun in Georgia and carry it with you as a California resident. This is due to California's laws, not Georgia's.
Texas, the state of your residency
Depends on Texas law.
From a DEALER- yes. From a private sale- no. A private sale may only take place in your home state between residents of that state. This is addressed in FEDERAL law and applies to all states.
Yes, if you live in Georgia.
The only way to do it legally is to have the Georgia resident send it to an FFL dealer in California, then take possession of it through that dealer. Georgia's definition of gun rights and California's do not go hand-in-hand, at all.