Yes, big Apple are compatible with the original Birdy fenders, the Birdy Rohloff is factory equipped with Big Apple. For the Birdy Touring, you have to remove two plastic washers mounted between the fender and the fork (these washers aren't mounted on the Birdy Rohloff).

Nice carbon wheelset !

For the airline size, no the folded Brompton doesn't fit into the allowed size but many airlines do not care. I have for instance transported my Brompton in the Radical Design Chubby trailer without any problem while this trailer is too big. The Birdy also fits in this trailer.