The remaining gap is smaller than the margin of error, once you account for every known factor. For example, a man and woman might both have the same job title at the same company, but if the man was working there for a longer period of time, or opts to work more overtime, etc. etc., then naturally he's going to get paid more "for the same work". But about that phrase:
You should understand that, primarily because it'd be absurdly impractical otherwise (no one is going to be examining the individual daily acts of all these people at their jobs), whenever research in this area talks about "same work", they always mean the same job title. So already, that's leaving a lot on the table, of which I gave two examples above (experience and amount of hours/overtime worked).
Fields dominated by men happen to pay more? Why?
You have cause and effect backwards. The fields pay more first, then men are shown to gravitate more toward them. This is partly because men tend to be more likely to prioritize raw earning potential over everything else, versus women, who are more likely to prioritize other things, such as time flexibility/convenience (check out the man/woman ratio of graveyard shift jobs for an eye-opener), commute time, etc. And part of the reason for that is the social pressure for men to be 'the provider', which may have lessened in recent decades, but is definitely still a factor to a degree.
Another big factor is that, as men are more likely to prefer 'working with things', and women are more likely to prefer 'working with people', the inescapable fact that 'things' scale up to a degree of magnitude that 'people' never can, means that the industries that men already tend to favor (STEM), will also be the ones that can scale up and pay more as a result of that. An engineer could be able to manage 1 system now, but be able to manage 10 in the future with technological advances, but even the best nurse on the planet is never going to be able to care for orders of magnitude more people than they can presently.
Why are men socially in a position where they can choose these higher risk occupations?
This is a loaded question. Men aren't any more "socially in a position" to do so than women. Women are completely free to choose these occupations. But by and large, they simply don't. The difference in priority I described above is why. Left to make a free choice, men are simply more likely to risk their safety and lives for a bigger paycheck, than women are.
Construction, oil field workers, logging all seem to have a pretty bad reputation for hostile work environments for women, no?
Okay, really now, let's not pretend there are these throngs of women clamoring to be 'let in' to the roofing industry, or the oil fields, and only aren't working in those fields because of the misogyny of the existing workforce. Please, let's return to reality here.
Why are men more often in positions where this is possible? What gender difference could there possibly be that could make this the case? What sexual dimorphism has led to this difference? What social expectations have we placed upon women that would lead to this? Personally I haven’t a clue.
Again, it's choice, not a difference in opportunity. I'm not sure why you're so hung up on that. Left to their own devices, and given full freedom to choose their professional paths, men and women, by and large, do NOT make the same decisions. In fact, the data has shown that the more egalitarian a society is re sex equality, the more pronounced those differences become (for example, the male skew in engineering tilts harder toward male, and the female skew in nursing tilts harder toward female). This is the opposite of what those who did this research expected to discover, such that it's literally called the "gender equality paradox".
Why would higher risks lead to a higher median salary?
Because if you have two jobs that have equivalent pay and prerequisites, but one is more dangerous than the other, no one will choose it over the safer option, obviously. You have to pay more for dangerous jobs, or no one will do them, unless they literally have no other choice.
9 . Why?
This is the 'working with things' vs. 'working with people' general preference difference between men and women, in action.
Why are men more able to relocate than women?
Once more, you're twisting things. Point 17 doesn't say men are more ABLE, it says they're more WILLING. Difference.