Istanbul Atatürk (TAV) airport is Europe, Sabiha Gökçen (SAW) is in Asia. The border is the straight of Bosporus.
Continents are a rather vague concept[1]. Any definition is going to be arbitrary. For the purposes of your contest, I think the spirit would be to have gone to the continents "proper", i.e. I wouldn't count Guam at all, but say the continents are the mainlands of: NA, SA, Africa, Europe, Asia, Australia, Antarctica. I'd be inclined to allow New Zealand as part of Australia as the only exception.
Other notable exceptions I can think of are Indonesia, Vancouver island and Manhattan.
I think you should extend each land mass to wherever the sea floor reaches a given depth (100 meters or so?), and then do a transitive closure on those to get continent boundaries.
Sure - why not, there's only a sea between us that you can fit an area the size of California to Nebraska into ... pretty close really.
Did I mention that any definition of continents is arbitrary?
We had a few issues in the beginning defining these border countries and decided to allow Istanbul as Europe, but will definitely group it for Asia in future.
We had a handful of difficult decisions, but luckily the winning entry visited all continents without contention.
Guam was the tough one for us. Would you say Oceania?