Try reading this site for bike-on-plane info, as well as comments on individual airlines: http://www.bikeaccess.net/BikeAccess/default.cfm The touring section might be worth a shot too.

I'm a proponent of just wheeling the bike straight on with pedals removed and the handlebar twisted. It causes the handlers to treat it like a bike instead of a giant box. You could try wrapping it all over with bubble wrap to prevent scratches. A downside is that some airlines require a box, or won't accept any liability without a box. The second point is a bit moot though as I don't think airlines offer much in the way of compensation for damaged luggage.

A cardboard box offers little protection (the one time I've used a box something was broken, vs the 15-20 times it was unboxed), and even if you pack it well it could still end up at the bottom of a pile of heavy things.

Other options- buy a used hard case and sell it after the trip. Get insurance specifically for the bike.

Another thing: I doubt that security is going to let you on board with a bunch of bike components (handlebars, seatpost) as hand luggage.