I don't trust TSA or baggage handlers with much. Obviously, you have no choice with your gun. I put my binoculars, GPS(if carried), and range finder in a back pack I carry on. Same with my pot calls, box calls, strikers and even mouth calls. I had a ceramic call broken when I packed it in my checked baggage years ago. Nevermore.
This way, I know I will have my calls with me and not messed up.
I usually wear a pair of boots and pack another pair.
I do put my knife, clippers and Leatherman in the gun case.
I place my shells in my checked luggage, not in the gun case, even though it's legal to do so. I've had issues before with the airline folks on this issue as they don't know any better. It's best to keep the shells in the original box to avoid aggravation.
As others have said, definitely place a lock on every place on your case that a lock can be placed. The airline folks are rigid on this even though a doubly locked case is just as secure as one with 4 locks.
Strikers as a carry on has never been an issue, no more than carrying a BIC pen on board which definitely could be used as a weapon.