Your understanding is largely right, though I'd expand "look like one gender" to "be perceived and treated as one gender". I regularly confuse the hell out of people with my obviously feminine name and appearance and very deep voice.
> Would gender dysphoria be less of an issue in a society that would not have such gender-assigned roles? Open question for me.
Depends on the person, I think. Many trans people still would. I probably would've just shaved a bunch of body hair off and started wearing skirts and been done with it.
As for pronouns, "he" is just another sound people use to refer to other people. It has no inherent meaning beyond what we ascribe to it. I don't use it because society does ascribe a lot of meaning to it that doesn't apply to me - the same sort of feeling a man would get from being called "she". If genders weren't so much of a thing in society, I can't imagine myself caring nearly as much, though maybe in that world we'd all use the same pronoun anyway.
But there's a whole spectrum of trans people - in the same way you don't really grok dysphoria, I don't really grok e.g. genital dysphoria, so I can't and don't intend to speak for anyone else here.