Variable versus fixed costs in the airline industry
I’ll address your second question first.

I'm fundamentally confused about the distinction between fixed and variable costs.

Simply put, variable costs are ones that go away if you cancel a flight (or, more likely, a pair of out-and-back flights) while fixed costs are ones you still have to pay.
A challenge is that many “fixed” costs are actually step-variable costs. For instance, if you cancel enough flights, you might be able to reduce how many aircraft or gates you need. But with the black and white dichotomy given, we classify those as fixed.

Landing fees (ATC, runways, taxiways, parking at stand or apron for a certain time (e.g. 2 hours), use of gates / fingers / terminal for disembarking).
Surcharges or rebates in addition of landing fees (for noise, peak-hour operations)
Passenger charges
Overflight fees

If you don’t land, there is no landing fee, so that’s variable. Ditto for gate fees, noise fees, peak-hour fees, passenger fees and overflight fees.
Space to park an unused plane isn’t usually an issue when it’s just one or two flights canceled; that will be included in some other fixed cost.

Lounges fees.
Scheduling coordination.
Administrative cost (managing/issuing tickets, etc.)

These are fixed (or step-variable), since canceling one or two flights won’t meaningfully change them.