Yes.
Yes the background check for guns is the same in every state
No
Yes- but only from a dealer. You may not buy a firearm from a private seller in ANY state other than your own.
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.