1. I came across the following: “When conversing with women, men will remove their hat.”
a. Is ”hat” correct here, or should it be replaced with “hats”? This seems to cause a lot of questions. Because it is clear that any one man will be wearing only one hat, and no two men can be weaing the same hat at the same time, there really can't be any confusion, regardless of which you choose. I would use the singular hat because in the end, we're talking about one man (and his one hat) at any given time.
b. What would the equivalent sentence in the singular sound: ”When conversing with a woman, a man will remove his hat”? That's exactly how I would do it.
2. I found the following combinations: “solar-powered”, “human-powered”, “nuclear-powered”, and “electric-powered”, “oar-powered”. I also found “electricity-powered”. Which of these forms are correct? You could do a Google search, but I'm far more familiar with "electric-powered" than "electricity-powered."
What rule, if any, should I apply : adjective+powered, or noun+powered ? Noun-powered, with the exection of electric. Perhaps electric simply stands in for "electrical energy" and it's just a shorthand used by convention.
3. Which of the following is to be found in British English:”car-rental station” or “car-hire company”? I'm American. We would say "car rental lot" (no hyphen), so we'll wait for a BrE to confirm.
4. Is the comma OK in : “On moving out, they threw a huge all-night party.”? Yes, I would say it's not only okay, but required. I would also put one after "huge."