What happens if you're flying to Europe with a bike but are flying to
the departure point? Will you be charged for the US flight if you're
on the same carrier overseas?

I have had the bicycle flown for free in that situation. However, it
has taken a conversation with the gate agent, along lines of "Isn't
this an international flight?" Also, on return may need to pick up
the bike, carry through customs and redeliver to US domestic transfer.

I generally try to be polite (e.g. not impatient, demanding, allowing
lots of time), but also politely asking or even being surprised at the
large fees. That has helped me in a number of situations where I've
been given some leeway).

--mev, Mike Vermeulen