Some of what people do varies depending on where they are. I see lots of pictures and comments online of europeans just wrapping their bikes in plastic or bubble wrap. Here in the US most of the bikes I see flying are in cardboard boxes from bike shops (the boxes bikes ship in when new), but some buy dedicated cases (hard or soft ones are available). Asking at a bike shop will usually get you a free box.

People routinely fly with their bikes for touring, but they also ship their bikes via ups fedex or whatever. There are services that help cut the cost a bit if they serve where you are (shipbikes.com or bikeflights.com).

I have sometimes paid a bike shop to pack and ship my bike for me. I personally only do that when returning home from a tour when I don't want to bother with the bike. Bike shops seem to get a lot better shipping rate than I do so it helps make up for the cost of them packing the bike. Mostly I do it to avoid dealing with getting a box and schlepping stuff around in a strange town when I am done touring ready to just go home though.

How much disassembly? Enough to get it in the box. That will vary with the size of the frame and the size of the box. Looking at the online how to articles is a good idea, but you don't need to go crazy. Some people get a little carried away.

Airline regulations may vary depending on both airline and where you are. You might be able to use a box like the ones we use on Amtrak trains here where you can pretty much turn the bars put it in the box and tape it shut. Adding a little padding is still a good idea, but no real disassembly is really required.