Hello David - As for riding away from the airport, I have done this a couple of ways. I don't want to transport a rigid suitcase on the bike or pull a trailer, so I use soft case that can be folded or rolled to a size easily transported on the bike.
1) I have placed the bike in a translucent Ikea bag (I think it is called a Dimpa bag or something similar) and gate checked it. After retrieving the bike at the destination, the Dimpa bag is easily folded and stashed in the Brompton t-bag or secured on top by the adjustable strap.
2) I have used the Brompton B-bag, gate checked or checked at ticketing, to which I add a protective layer comprising a thermo-rest z-lite sol folding sleeping pad for extra protection. The pad is cut to provide a layer beneath the bike and the remainder wrapped around the bike and the B-bag zipped around it. The pad folds accordion style to fit in the bottom of the bag and when the bag is folded over it takes up no more or very little more room than without the pad (I open it to double wide / half height before placing it in the bottom of the bag). The B-bag is then placed on top of the T-bag and secured with the single adjustable strap. I just got back from 2-1/2 weeks in Spain and during one stretch travelled this way for a few days on a portion of the Camino de Santiago and through Rioja and the arrangement worked great.
As some have said, when it comes to riding out of airports, some are easier than others. I had read of people riding out of the Barcelona airport but realized too late that they must have landed at the other terminal. We gave it a go and realized after about 1/2 hour of false starts that we couldn't avoid getting on a highway for a good stretch with little or no shoulder. So back to where we came from to hop on the aerobus for a 1/2 ride to the heart of town and a 5 minute ride from our hotel. The joke was on me as after landing I was pointing out the aerobus - about 50 feet away - to my wife as we were unpacking the bikes and getting ready to ride, only to return to it later. But it is nice to have the option and in 30 seconds were off the bikes and ready to board the bus.
FWIW, the Spain trip was a great success. We travelled by international airlines, regional airlines, buses (great system in Spain), train, light rail, taxi and our own leg power to get from city to city. Even took the Bromptons to the top of hills via funicular in Barcelona and San Sebastian and cruised down (amazing in Barca coming down from Montjuic Castle to see the Olympic site, radio tower and an otherworldy cemetery). Barcelona is an incredible place to bring the Bromptons and must be one of the great travel destinations for Brompton owners - they just greatly enhance the travel experience in such a bike friendly city. Don't know if it's true but I heard that 10% of Bromptons sold are sold in Barcelona.