The basic answer is Yes .a firearm can be sent through the mail, but obviously all the relevant documentations need to be in order and I would be surprised if you had to use a specialist courier. Also if the gun is registered to you, then title will also have to be transferred to your father.
The only way legally to mail a firearm is to mail it to your self to a forwarding address no ammo in the gun of course other wise you need to go to a gun dealer to mail it to another gun dealer ffl to ffl don't play with this it can get you in trouble call the atf for more info
Yeah you can