It's been my experience that the TSA is more concerned with checking the padding in my gun case for contraband than they are concerned about the gun itself--they just ask me if it's unloaded, I say yes, that's that. Also, different airports handle it differently. Smaller airports take it out, have you unlock it there (usually in a back room), manually check it (and the rest of your luggage) and then shunt your luggage to the "head of the line" as it were. Kind of nice that I get to be there for my luggage inspection. The bigger airports which have the sophisticated x-ray machines just want to know about the existence of the gun in there so they don't freak out when seeing it on x-ray. They ask you to hang around while it's scanned, none of them have ever asked to open the luggage.
Factory box for the ammo. I have travelled so much my box was starting to get worn so I wrapped it a few times with clear packing tape. Now I have a sturdy factory "travel" box.
I also make sure to put my cell phone number at least on my luggage tag if not on the gun case itself.
I'm astonished that it gets no special treatment when dumped on the baggage carousel. I was at an airport once where all the flights were cancelled due to weather. They just dumped my bag on the carousel and about 4 HOURS later when they finally admitted I wasn't going anywhere I went out to find that my bag, with the gun in it, had been on the carousel alone the whole time--anyone could have taken it.