What's wrong with "jechać"? Especially considering that people VERY rarely walk to the airport...
Ambivalent - "udaje się" would be the most appropriate translation of "going to" in my opinion, while "idzie" would be rather an uncommon way of reaching an airport.
Is there a rule that explains why the preposition here is "na"? Would "do lotniska" be possible?
The rule of thumb is that 'na' is used with open spaces, but there are of course exceptions from that.
'do lotniska' doesn't work here.
Why not "idzie"? Does it imply walking?
Yes, this implies walking, which obviously is technically possible. "idzie" is an accepted option, it should have worked.