Does anyone know which fees the airline gets to keep? I'm guessing they keep the $350 and possibly the last two.

Personally, I think that only airport/government fees/taxes should be listed. Money that goes to the airline (including fuel charges) should just be included in the fare. It is absurd to charge $1 for a trans-Atlantic flight. Just say it's $351. I'd imagine part of the appeal is making the fee seem mandatory, instead of an optional airline-imposed charge. Or perhaps some fees are based on percentages of the 'fare'.