"My dear madam, I am not so ignorant of young ladies' ways as you wish to believe me; it is this delightful habit of journalizing which largely contributes to form the easy style of writing for which ladies are so generally celebrated. Every body allows that the talent of writing agreeable letters is particularly female." (3.27)
Henry is teasing Catherine about the cliché that all young ladies keep journals and write nice letters. Henry is very funny here, as he ridicules gender clichés.
"I should no more lay it down as a general rule that women write better letters than men, than that they sing better duets, or draw better landscapes. In every power, of which taste is the foundation, excellence is pretty fairly divided between the sexes." (3.33)
Henry seems quite liberal in his opinions on gender here, though he does qualify his statement of equality by linking it to matters of taste. Interestingly, Henry lists the abilities in which Catherine is rather deficient – so says the narrator at least.
for if it be true, as a celebrated writer has maintained, that no young lady can be justified in falling in love before the gentleman's love is declared, it must be very improper that a young lady should dream of a gentleman before the gentleman is first known to have dreamt of her. (3.51)
Gender expectations even govern how men and women can behave in matters of love and romance here.
[…] Catherine, after listening and agreeing as long as she could, with all the civility and deference of the youthful female mind, fearful of hazarding an opinion of its own in opposition to that of a self-assured man. (7.32)
Catherine is silenced and subdued by the overbearing John Thorpe here. Her youth is acting together with expectations for women's behavior to make her timid around John.
Little as Catherine was in the habit of judging for herself, and unfixed as were her general notions of what men ought to be, she could not entirely repress a doubt, while she bore with the effusions of his endless conceit, of his being altogether completely agreeable. It was a bold surmise. (9.32)
Catherine has the double problem of not being confident enough to express her opinions, especially around men, and not being confident enough to have formed opinions worth expressing.
It would be mortifying to the feelings of many ladies, could they be made to understand how little the heart of man is affected by what is costly or new in their attire […] Woman is fine for her own satisfaction alone. No man will admire her the more, no woman will like her the better for it. (10.21)
The narrator humorously highlights communication problems between the sexes in terms of fashion, where many women are convinced men notice what they wear.
"In marriage, the man is supposed to provide for the support of the woman; the woman to make the home agreeable for the man [....] But in dancing, their duties are exactly changed; the agreeableness, the compliance are expected from him, while she furnishes the fan and the lavender water." (10.34)
Henry jokingly compares marriage and dancing here, and comments on gender roles and gender expectations in polite society. Expectations for women are rather low and tend to revolve around pleasing men, while men are expected to care for women.
She was heartily ashamed of her ignorance. A misplaced shame. Where people wish to attach, they should always be ignorant. To come with a well-informed mind, is to come with an inability of administering to the vanity of others, which a sensible person would always wish to avoid. A woman especially, if she have the misfortune of knowing anything, should conceal it was well as she can. (14.27)
The narrator is satirizing, or ridiculing, romantic relationships and social expectations for women here. Women are supposed to be many things in this society, but well-informed isn't one of them.
"Miss Morland, no one can think more highly of the understanding of women than I do. In my opinion, nature has given them so much, that they never find it necessary to use more than half." (14.46)
Henry jokes about his feelings on women's intelligence here. As a rich man, Henry likely had many women targeting him for his money, and behaving ridiculously while doing it.
"But," said Eleanor, after a short pause, "would it be to promote his happiness, to enable him to marry such a girl? - She must be an unprincipled one, or she could not have used your brother so." (25.30)
Eleanor is speaking of Isabella and her relationship with Captain Tilney here. Isabella's reputation is pretty much ruined. Though Eleanor is related to the Captain, it is notable that she comments on Isabella's reputation and behavior and not so much on his.
He went; and, it being at any time a much simpler operation to Catherine to doubt her own judgment than Henry's, she was very soon obliged to give him credit for being right, however disagreeable to her his going. (26.15)
Catherine definitely hero-worships Henry, and frequently bows to his judgment.