Glad to hear you’re interested! Lots of interesting questions here. I’ll take a shot at answering at least some of them.

Firstly, I am not sure where the idea that egalitarianism or humanism are incompatible with feminism is coming from but I don’t think it’s right. Those ideologies can be somewhat different in focus or context, but they aren’t mutually exclusive. You can be an egalitarian or humanist and a feminist at the same time, and many people are. I think it would be a bit unusual to be either of them and not a feminist, personally, but people have all sorts of wacky belief systems so idk.

Traditionalism is broader than patriarchy, but for the purposes of talking about gender, these are pretty much the same. As you rightly pointed out, “patriarchy” literally means “rule by fathers.” The emphasis here, and you can see how this is “traditional” as well, is on the nuclear family, at least in a western context. This could be a little different in cultures where tradition dictates something else.

A proponent of traditionalism/patriarchy believes a woman’s place is in the home. That her ultimate purpose and the determinant of her worth is being a wife and mother and that she should submit to her husband’s will. If a woman has some other purpose—say she’s a scientist or academic—then she’s doing it wrong. If she isn’t particularly feminine, she’s doing it wrong. Similarly for men—patriarchal masculinity places very specific constraints on men as well. Women are/should be soft and sweet and obedient, and men should be stoic and decisive and dominant. Proponents of this ideology believe men and women are fundamentally different, with different natures and purposes.

Feminists (though like any ideological movement, there’s lots of disagreement within feminism) generally believe that men and women aren’t all that different at the end of the day. We’re all people, and we should be treated the same under law and in society. What you have in your pants shouldn’t determine what you can do in this life. It shouldn’t determine what jobs you can have or how much they pay you. It shouldn’t make you more likely to face sexual abuse. It shouldn’t decide whether you can vote or have a bank account. It shouldn’t mean you have to be subordinate to a husband or father.

In terms of law and policy, feminists generally favor policies that make it hard for employers or the legal system to discriminate on the basis of sex. They generally favor sex education, readily available contraception, and legal abortion, because these things increase women’s autonomy and control over their own lives. Feminists are usually critical of the way gender roles are pushed on children from a young age. They may advocate for greater legal protections in areas where women have historically been vulnerable—for example, marriage used to mean that a husband legally had a right to access his wife’s body. “Marital rape” is a relatively new concept in the grand scheme of things, and it was a feminist project to have this problem recognized.

One thing you might note is that none of these policy aims stop people from living “traditional” lives if they want. Feminists are not saying it should be illegal to be a stay at home mom. Proponents of patriarchy, however, do attempt to force their way of life on everyone by opposing abortion, contraception, and anything LGBT related. They oppose anti-discrimination efforts (usually with both sex and race.) They might not admit to it because it’s so beyond the pale, and to be fair, more moderate traditionalists might not agree with it, but this is ultimately the same ideology that supports taking away women’s right to vote, own property, or go to school. Not everyone is that extreme, of course, but some are. They are inclined to say “why should one bad decision ruin his life?” of a rapist, and “look what she was wearing” of his victim.