Getting a Brompton on and off a plane without going broke
You get what you pay for - you cannot travel heavy (A bike is traveling heavy) and cheap. Traditionally US has the highest baggage allowances in the world, international travel outside the US (cattle class), a baggage allowance of one 20kg bag is common. Excess baggage is often calculated at 1% of a first class airfare per kg or something similar. Most airlines have concessions for sports gear. 

This is the main reason why there are still Travel agents. They should know this stuff and have access to much better travel planning tools than you do. Their business model today is stepping in when google (an patience ) fails. 

Second part is shop (and you get what you pay for) better travel insurance, and read the policy before you need to claim. You seem overly concerned about getting your bike damaged or lost, to the point its stopping you in your tracks and you are not enjoying the travel experience (planning being a part of that).