That Lady Russell, of steady age and character, and extremely well provided for, should have no thought of a second marriage, needs no apology to the public, which is rather apt to be unreasonably discontented when a woman does marry again, than when she does not; but Sir Walter's continuing in singleness requires explanation. (1.9)
This underscores the different roles of men and women in marriage during this period. Lady Russell's got her financial stability, so she no longer needs a husband, while a widowed man of good fortune must be in want of a wife to keep his house running for him.
The child was to be kept in bed and amused as quietly as possible; but what was there for a father to do? This was quite a female case, and it would be highly absurd in him, who could be of no use at home, to shut himself up. (7.7)
So daddy doesn't know any quiet games? This sounds like an excuse...
[Mary speaks] "So you and I are to be left to shift by ourselves, with this poor sick child; and not a creature coming near us all the evening! I knew how it would be. This is always my luck. If there is anything disagreeable going on men are always sure to get out of it, and Charles is as bad as any of them." [...] [Anne speaks] "Nursing does not belong to a man; it is not his province. A sick child is always the mother's property: her own feelings generally make it so." (7.10)
And Mary thinks that Charles is being slippery too. Anne here is the voice of the status quo, and gets in a subtle dig at Mary: if she were a good mother, she would want to stay with her child.
"Altered beyond his knowledge." Anne fully submitted, in silent, deep mortification. Doubtless it was so, and she could take no revenge, for he was not altered, or not for the worse. She had already acknowledged it to herself, and she could not think differently, let him think of her as he would. No: the years which had destroyed her youth and bloom had only given him a more glowing, manly, open look, in no respect lessening his personal advantages. She had seen the same Frederick Wentworth. (7.34)
Just like any movie where a 50 (or 60, or 70)-something -year-old guy is paired with a 20-something girl, the aging process is viewed very differently, depending on whether the person getting older is male or female. Compare, too, Sir Walter's views on beauty: he seems to think himself still handsome, while he criticizes Anne and Mary for looking old.
[Captain Wentworth speaks] "Such a number of women and children have no right to be comfortable on board." [...] [Mrs. Croft speaks] "But I hate to hear you talking so like a fine gentleman, and as if women were all fine ladies, instead of rational creatures. We none of us expect to be in smooth water all our days." (8.45)
Captain Wentworth's comment here is kind of like what Anne says about good mothers liking to take care of their sick children: Wentworth thinks women should be uncomfortable on board ship, and no amount of actually quite comfortable women will make him change his mind. Mrs. Croft points out another fallacy in her brother's argument: that he's taking a small number of fussy women and pretending all women feel that way.
"And yet," said Anne to herself, as they now moved forward to meet the party, "he has not, perhaps, a more sorrowing heart than I have. I cannot believe his prospects so blighted for ever. He is younger than I am; younger in feeling, if not in fact; younger as a man. He will rally again, and be happy with another." (11.13)
Here's that gendered age difference again, that turned up before when Anne compared the change in her looks to Wentworth's; a man is considered young for much longer than a woman, and so has the opportunities of youth for longer.
[Anne speaks] "We certainly do not forget you as soon as you forget us. It is, perhaps, our fate rather than our merit. We cannot help ourselves. We live at home, quiet, confined, and our feelings prey upon us. You are forced on exertion. You have always a profession, pursuits, business of some sort or other, to take you back into the world immediately, and continual occupation and change soon weaken impressions." (23.19)
Anne suggests that differences between women's and men's feelings are in part due to their different social situations – men can go out and distract themselves in a way that women (good women, anyway) can't.
[Captain Harville speaks] "But let me observe that all histories are against you--all stories, prose and verse. If I had such a memory as Benwick, I could bring you fifty quotations in a moment on my side the argument, and I do not think I ever opened a book in my life which had not something to say upon woman's inconstancy. Songs and proverbs, all talk of woman's fickleness. But perhaps you will say, these were all written by men." (23.27)
See what happens there? Harville says that nearly all books that represent female characters do so from a male perspective – by saying what he thinks Anne is thinking (so, putting words into her mouth from a male perspective).
[Anne speaks ] "Yes, yes, if you please, no reference to examples in books. Men have had every advantage of us in telling their own story. Education has been theirs in so much higher a degree; the pen has been in their hands. I will not allow books to prove anything." (23.28)
Anne agrees with Harville's idea of her position, though – suggesting that perhaps not all of one sex's ideas about the other are wrong. Anne connects the single-sidedness of literature with larger issues of female education, directly linking the gender imbalance in literature to gender inequality in society.
[Anne speaks] "I believe you [men] equal to every important exertion, and to every domestic forbearance, so long as--if I may be allowed the expression--so long as you have an object. I mean while the woman you love lives, and lives for you. All the privilege I claim for my own sex (it is not a very enviable one; you need not covet it), is that of loving longest, when existence or when hope is gone." (23.32)
While Anne is making a statement about gender difference here, she's also very conscious that Wentworth is eavesdropping on her conversation – how much influence does that have on the positions she takes?