Consider purchasing a hard sided bike case. We've taken my husband's bike with us as checked luggage to multiple places around the world. We would never consider packing it in a cardboard box. I've seen what those look like when they bring the oversize luggage out.

As far as a waiver, we've never had to sign one on United or BA, but even without it, you'll be limited to the airline's luggage reimbursement policy.