As others have said, check out your local bike shops. Timing matters here. Most bike shops spend the weekends with sales and repairs, and then early in the week, they build bikes. So, if you check with them on Thursday or Friday or a weekend, they have probably already tossed/recycled the boxes. You've got a better shot on a Monday, Tuesday or Wednesday.
Get as much of the associated packing materials as you can, especially the little plastic piece that fits in between the fork blades at the dropout.
I just shipped a bike that I sold. It was pricey, and I was fond of the bike, so I wanted to make sure it made it to the new owner in great condition.
I mostly followed this method:
https://www.rivbike.com/pages/how-we-pack-your-bikeGenerally, what you want to do is to wrap everything up so that things are not moving around and banging into things, and also make sure that potential sharp points won't poke through the box. So pay particular attention to the fork ends and where the seatpost goes into the frame. Bikeflights also has some packing hints on their website.
Bike tires won't blow in an airplane hold- the difference in ambient pressure between ground and the cargo hold doesn't come to much in terms of psi.
I would highly recommend bikeflights.com. You don't need to buy the box from them to ship with them, but you will need a few photos to document how you packed the bike (if there happens to be a claim), so I'd suggest this order:
- gather materials (bike box, pipe foam, packing tape, newspaper works pretty well)
-measure box
- set up shipment with bikeflights.com
-pack bike, taking a few photos as you go
-get final weight and then set up a pickup via bikeflights
Hope this helps!