To keep weight down, put stuff in your other luggage. Shoes, pedals, helmet, tools, lights etc. If you're just over weight, remove the chain and pack it in your bags.
TSA will open your box. Make it easy for them to open, pack up correctly, and close. Having stuff out makes that easier.
Actual airline charges are all over the place. I've been charged $100, $200, 100 Euros and 0 on international flights. Prepare for the worst, be nice to the agents, go with the flow.