BudsGunShop.com - Shipping

Joshua, no Sir. The buyer would have to be the registered owner. You can purchase your dad a Buds Gift Card in the total purchase amount and he can purchase it himself : budsgunshop.com/search.php/type/Gift+Cards

NO, that is NOT legal.

Unfortunately no, even if you were in the same state, the purchaser must be the same person to pick up the fire arm from the ffl.
What you can do is you can purchase a gift card in the amount of the firearm and email them, or print and mail them and have your dad purchase the gun in his name.
This isn't just a buds rule it is a law by the ATF.
Gift cards available here
budsgunshop.com/search.php/type/Gift+Cards

No you can not, your best bet is to get your dad a gift card and let him make the purchase himself.