No comment on the Bike Friday, but I actually found traveling with my road bike to be very very easy. I posted a thread on it a few weeks back asking about airline cases and I ended up going with the Biknd from Canada. The total weight of my bike and the bag was less than 50 pounds and I wasn't charged an oversize bag fee. Because it was special, it came off the plane first so I didn't have to wait around in baggage claim either. Admittedly, the case is kind of pricey, but for me it was totally worth it to have my own bike with me that is fit to me with all my things. The case had plenty of room for my helmet and shoes and I bet you could get a lightweight pump in there too (since no CO2 cartridges on the plane).

EDIT to add: Thought I should also mention that it fit in the trunk of my dad's sedan without a problem at my destination.