I doubt very much that FedEx or UPS do that. It would create a liability for them if they fail to detect some dangerous content. It would also destroy privacy of shippers - and that is something that courier services cannot afford. X-ray may also damage shipped materials, such as exposed but undeveloped film, medical samples, some fine electronics. There is simply no upside for FedEx to bother with X-ray; it's a lot of expense and no profit.
FedEx will not legally ship a firearm from you to your house. One of the ends must be a licensed dealer. Interestingly enough, the law does not seem to allow this particular situation where a lawful owner ships his legally acquired firearm from one place where he can have it to another place where he can have it.
You must disclose that the package contains a firearm; then the package will be labeled on the outside (so that it can be conveniently stolen by low wage workers, I guess.) It would be a violation of the contract between you and FedEx (and of the state/federal laws) to put a firearm into a metal box, wrapped well so that it doesn't rattle, and ship that. Though the risk of discovery is very low, if the matter becomes known the punishment might be very unpleasant.
If I were to be caught in this situation, I would have called a dealer in my area (a few are well known to me since I, naturally, visit their places of business now and then) and asked them to receive and hold the firearm for me. They'd charge a nominal fee for bother, about $25 (the same they charge when you buy a firearm from someone else and have it shipped.) That would be 100% legal. Driving is also perfectly legal as long as the firearm is unloaded and in a locked container in the trunk. That's what hunters do all the time. If some cities ban firearms in every way possible, steer around them (they are likely to be bad places to visit anyway.)
“X-ray” is used in my comment as shorthand for the various scanning technologies in use. Gamma scanning, laser-based molecular scanning (more for explosives but can identify gunpowder), millimeter wave and others. All of these can and have been installed in carrier properties on a temporary or permanent basis in response to perceived threats. The shippers have no choice but to be extremely accommodating to DHS when the request comes in.