Anyone can ship a rifle, shotgun or handgun across a state line to an FFL dealer, and anyone can ship a C&R across a state line to a C&R (collectors) FFL holder. But arrangements should be made with the dealer so he expects the shipment, and you will need a signed copy of his FFL. Some dealers will not accept shipments from non-FFL holders under any circumstances and that is their prerogative.
As to taking gun on plane, check with the airline. Target shooters take their guns all the time and most airlines accommodate them. Some airlines will no do so.
Alternatives are driving, with gun unloaded and locked in trunk.
As for other shippers, like FedEx, ask them. If you are going to ship on a company account, better clear with the company first (unless you own the company), as some outfits consider misuse of their shipping accounts a firing (pun intended) offense.
Jim