You ask about boxing your bike at the airport. Unless you plan to ride your bike to the airport carrying a box, it sounds like you are expecting and counting on the airline to supply a box. I would not go that route unless there is a 100% chance that a box will be available. A safer bet is to supply your own box. If you go that route you might as well box it up before you get to the airport.
Asking others what airlines they have used will only elicit helpful answers if someone happens to have had an experience with an airline that flies to where you want to go. I will be flying to MSO in June. If I asked the question you did and someone responded "I used Southwest. They are the cheapest for bikes.", the response would not help me because Southwest does not fly to MSO.
All that aside, if your travel plans are within the U.S., have you considered shipping your bike? I will be shipping to Missoula this June. I have used bikeflights.com on two occasions, saving A LOT of money over what I would have had to pay the airlines. Last year I shipped from Philly to Rapid City, SD for something like $47, including a 10% discount for repeat business and $1,400 in insurance. I think my airline wanted $150 for the bike. For what I saved, I had the bike professionally boxed for shipment and assembled and tuned by a local bike shop in SD. And there is no dealing with lugging a bike to the airport, going through security and having a TSA agent open the box and not close it properly. One time I flew with my bike the TSA agent couldn't figure out how to use the buckles on my box's straps. When I got to my destination, I found one of the straps tied in a crude knot.