Let your ticket agent know you are traveling with a firearm.
Follow these exactly. "Not accessible" means that. No TSA locks, I used Master locks and I have the only key. Do not yield the key to anyone unless you are present when/if they want to examine the gun (they will do this at checkin). Be sure it is unloaded!
Be sure to check that your baggage claim check is for the correct destination. A friend had his bags and gun sent to a different city by a "probable" bunny-hugger ticket agent.
Have a printed copy of the TSA rules and your airline's rules with you. Be polite, ask questions if you have concerns. I've never had a problem with domestic flights and only one problem with U.S. Customs when returning to the U.S. In that instance, the Customs agent got his butt in trouble. The ticket folks and TSA staff have always been friendly and professional with me.
Have a great hunt! Post a report on your experience.
I always put my shells in my check bag. Have never had an issue.
According to a TSA supervisor that my roommate knows personally, they can go in either your check bag, or your case.