FALSE, you may not ship second day. All carriers require firearms to go overnight. By law, you are required to inform the carrier in writing that you are shipping a firearm. So the only way to ship 2nd day is to not tell the carrier; a violation of law.
If you're a private citizen, you may ship a rifle or shotgun through the mails--you can send the thing 4th class for all anyone cares.
If you're an FFL you can also ship a hangdun through the mails and again, you may do so via any service you want to.
Contract carriers are a different ball of wax--they can set the rules to whatever they want to. IIRC, UPS (might be Fedex) will let you get away with shipping a gun via their 2-day air service, but you'll have to talk to your local manager to determine that for sure.
It's also important to note that there are gun friendly, gun unfriendly, and gun ignortant shipping personnel out there. Some in the former group will let you ship a gun via UPS ground, while some in the second group will refuse to accept your firearm for shipment at all. The ignorant usually fall into one of the two extremes.
Oh, and you are completely correct: The receiving dealer MUST provide a signed copy of the FFL to whoever is shipping the gun.