All of this is framed from a US perspective, I apologize to the extent that it's relevant.

getting paid less for the same work

Essentially not a real thing, and if it is happening at a particular employer it's illegal and time to sue. The wage gap that's published is measured as the difference in median total earnings for full time year round workers by sex, and any attempt to constrain it further to be "for the same work" (like adjusting for industry, role, hours worked, experience, etc) rapidly causes it to diminish. It is at it's heart an artifact of differences in the average life path of men and women - to the point that young, childless, urban, educated women actually earn more than similar men.

and not at all for all the work that isn’t considered work by men like raising kids and running a home

Taking care of one's home/family isn't paid work for anyone, regardless of sex. Men aren't paid for more stereotypically male housework either, like lawn maintenance, cleaning gutters, dealing with pests, plumbing or electrical, that sort of thing. If you do domestic work for another household, generally you do get paid for it.

Also, there's no third party mandating anything about how your household divides the tasks necessary to keep things going - you negotiate your own division of household labor with any partner(s) or roommate(s). For example in my household my wife and I both work full time, and for most "departments" of stuff that need done we each take a role. She does the laundry, I fold and put away (because her clothes have more complicated cleaning directions, and it's harder for her to lift and haul stuff around). Whoever cooked doesn't do dishes. I bring in groceries, she puts them up (the steps and heavy lifting are easier for me). Etc, etc.

again, statistically it’s pretty much all women

Dig deeper into those stats. Specifically, look at the differences in numbers that measure recent victimization versus longer periods. What you tend to see is the more "fresh" the experience is (looking at recent months or years rather than lifetime) the more likely men are to report it (almost as though men are repeatedly told by society that they can't be victims of sexual assault and doubly can't be the victim of a woman until they internalize it so they mentally file those experiences away as something else [if you can't be a victim then what happened can't be a violation]- I'm speaking from experience on that one) and previous 12 month numbers fare closer to like a 60/40 split presuming you don't also do some trickery of categorization where (for example) ways a woman are likely to sexually assault a man get filed into a subcategory of "other" to make the comparison less obvious, with women being a majority of perpetrators against men (ignoring the incarcerated of course because then men are a large majority of both perpetrators and victims - there's a reason term "rape culture" was originally coined to describe prison).