Very well summed up. I'll add that the counterperson may not be aware of current firearm regulations and get flustered, trying to call their supervisor and such. They may even argue with you, so have the airline regulations printed out beforehand.
The guy that handled my situation was probably new and assumed I was LEO and tried to hand me a form so I could carry a loaded firearm on the aircraft.
Patiently I informed him I was not LEO, and his demeanor slowly got jittery. The TSA agent took over from there.
It'll be somewhat embarassing to tell the agent you have a firearm, then wait for the TSA guy to check your equipment in front of everyone in line, but don't worry...happens all the time.
Oh, and my bag didn't come down the baggage claim carousel, but was set aside in a corner with other luggage. So if your bag doesn't show up, don't do like I did and wait 45 minutes only to discover it's somewhere else.