Summary- With very few exceptions (inheritance being one) when OWNERSHIP of a firearm is being transferred across a state line, the firearm must be shipped TO a FFL in the recipients' home state. When you are sending a gun TO a repair facility, and the same (or a replacement) gun is returned to you, there is no transfer of ownership. You can also ship a gun to yourself (going hunting in Montana, send your rifle to Big Sky Lodge, hold for guest C3shooter) Again, no transfer of ownership.
Shipping- a private citizen can MAIL a rifle or shotgun by US Mail. Again to a FFL is crossing state line, ownership being transferred. They CANNOT mail a handgun. The CAN ship a handgun by UPS or FedEx, which requires Next Day Air, adult sign, which is about $70-$75. You can take it to a FFL (dealer) who CAN use US Mail for a handgun. May be a lot cheaper,
For anyone that tells you "wink wink nod nod, just list it as machine parts, drop it in the mail"- yeah, sure. Know what Fed law enforcement agency has the highest conviction rate? Postal Inspection Service.