In her first passion Woman loves her lover, In all the others all she loves is Love, Which grows a habit she can ne'er get over, And fits her loosely—like an easy glove (3.3)
Byron makes a generalization in saying that all women love the first man they have sex with. But from that point on, they only love the idea of love. It's like the difference between loving a guy for who he is and loving someone because you want your life with him to be like a romance movie. The first is based on the person and the second is based on the idea of a relationship. We know, ladies. Throw your popcorn at the screen (or the nearest picture of Lord Byron) now.
Nor to the trouble which her fancies caused; Yet even her tyranny had such a grace, The women pardoned all except her face (5.113)
When Don Juan finds himself dressed as a woman in a sultan's palace, he spends the night with all the sultan's other female slaves. These women are okay with Don Juan except for the fact that he has such a pretty face. In other words, they find him to be a pretty woman, even though he's a dude.
Her sixth, to stab herself; her seventh, to sentence The lash to Baba: —but her grand resource Was to sit down again, and cry—of course (5.139)
The sultana is about to kill Don Juan because he has rejected her sexual advances. But just when her rage hits its peak, she breaks down and cries. When Byron says, "of course," he's playing into the stereotype that women are emotional and erratic.
The Turks do well to shut—at least, sometimes— The women up—because, in sad reality,
Their chastity in these unhappy climes
Is not a thing of that astringent quality (5.157)
Byron suggests that the Turkish people are right to treat their women as slaves because, in that part of the world, the hot climate makes women want to commit adultery. How's that for nineteenth-century logic?
I'm a philosopher; confound them all! Bills, beasts, and men, and—no! not womankind! With one good hearty curse I vent my gall (6.22)
Byron says that he wants to curse pretty much everything about him except women, although he might be sarcastic in his tone here. Most of his criticism he reserves for the shallowness of English political society and, for the most part, he treats women as innocent because they aren't allowed to participate in political society (at least not in Byron's day).
And next she gave her (I say her, because The gender still was epicene, at least In outward show, which is a saving cause) (6.58)
Byron excuses himself for referring to Don Juan as a "she" because he is dressed as a woman at this point in the poem. This clever use of the female pronoun reminds us just how easy it is to bend our ideas about gender being something that's fixed from birth.