My experience has been that regardless of advance planning and one's best attempts at nailing down the rules, it all comes down tho the overworked airline employee at the check-in counter. I have had the best luck when I have removed both wheels and organized the bike and wheels to fit in the smallest possible box. This is generally something on the order of a BMK sized bike box. Sorry I can't provide measurements, I throw 'em away when I'm done. My technique involves some disassembly but I don't remove the fork. I try to put it together so that the whole thing can be pulled out of the top of the box in one unit so TSA can eyeball it and slip it back in with minimal effort. The airline counter people aren't going to look in the box and unless the box says "bicycle", are not going to know or even question what's in the box. I usually construct, or reconstruct, a box out of more than one bike box, using a utility knife and a hot glue gun, making it as small as practical and "rolling my own" cardboard internal cross pieces to resist crushing. I also usually add a couple extra flat pieces of cardboard on each long side to make the package a little more rigid and puncture resistant. I cut a couple oblong hand-hold handles, one on each side close to opposite sides at the ends (wherever it seems to make a balanced carry) to make it easy to pick up and hopefully less likely to be abused by a frustrated baggage handler. So far this has worked out without any damage or baggage up charges. I have not, however, flown on Southwest. Do the best you can to make it small, don't label it a bike, and smile as you approach the counter.

Good luck and safe travels