> If I was a government official and decided to host a public event at a private property which I paid for, where the private property said no women allowed (say it was a private club which allowed to do such), could I use that rules to prevent women from attending the event?

Even if it were a public forum (which not all public events involving a government official are), I'm not sure that public forum law prohibits sex discrimination as well as viewpoint censorship. I can see an argument that it should (and that it shouldn't), but I don't recall. seeing any case law either way.