I think Zee is just generally dismissive and his manners leave a lot to be desired.
Whether this is due to sexism is another matter; it may be but I'd need more concrete evidence before making an absolute decision.
The thing is, everything that appears to show Zee as dismissive towards females (and there is evidence, no doubt about it), can possibly be explained another way.
The door incident was simple rudeness. Nothing to do with whether the person following him was male, female or Jason.
Choosing Natalie to bring back to the boardroom - as has been said, Zee was stuck a bit between a rock and a hard place with this one.
Kurt dropped a clanger with the flag measurements but again as people have stated, it wasn't a big mistake in context and certainly didn't cost them the task. Zee was also there when the mistake was made and did nothing to correct it.
Neil's mistake was bigger in context but I think Zee would be nervous of bringing Neil into the boardroom both from a past performance and an argumentative point of view.
The argument that Uzma tried to put forward - i.e. that she was an easy target, applied far more to Natalie. Natalie hasn't won a task yet, she lost badly as PM and she hasn't really done anything of note. I can see quite easily how Zee would feel she was an easy target to bring back to the boardroom; it backfired badly but it's hard to see how bringing back anyone else would have resulted in anything different.
Zee was dismissive of Natalie throughout the task but the reasons outlined above may be why. She hasn't been on the winning side or done anything particularly good. Yes, Zee appeared very dismissive and rude but it's not clear that he was like that because she's female. I think he'd be just the same to Jason for instance. If he doesn't rate you, he doesn't consult you - no tact, no diplomacy. It's not necessarily to do with gender bias. He's an arrogant, dismissive twonk but possibly not a sexist one.
I'm mainly playing devil's advocate here. I think it's quite likely that Zee has some difficulties accepting female authority. However, I don't think the evidence is conclusive.
Sexist? Very possibly.
Arrogant and dismissive? Almost certainly.
Fired? Incontrovertibly.
Move on