I've used a Cross Check (which is the same bike) for long distance and so far it's worked out pretty well.
It's a versatile frame, so you can use it for a wide array of terrain. Keep in mind it's not sold as a complete bike, so you will have to kit it out. But with the right type of rims, you can use quite a few types of tires on it. Keep in mind it won't work for serious MTB, but it's fine for dirt and gravel roads.
As to saving the fees, unfortunately it is no longer 2002. Many airlines are charging fees for bringing any luggage on the plane. In a lot of cases, they will not care as much about size as they will about weight; after all, the heavier an item is, the more fuel gets used. You'll have to frequently check airline policies for weight requirements.
By the way, I found packing and unpacking bikes to be a PITA; getting the bike to and from the airport is also a drag; and there is always the risk of damage in transit. And of course getting the couplers is adding around $1,000 to the price tag.
IMO it isn't really worth it unless you are going on tour for at least a week, and doing those tours more than once a year.