I don't think there's any sensible answer for most of your questions, but I can comment on the last one, at least. One of the main functions of a women's shelter is to offer living space and necessities to women who aren't able to be financially independent when they leave an abusive situation. Since the most common situation is that a man leaving an abusive situation is financially independent, there's little that a hypothetical men's shelter could offer him that wouldn't be adequately served by a cheap motel (or even a traditional homeless shelter, in the worst cases).

I think there have been a few efforts to open men's shelters, but to my knowledge none of them have operated for terribly long, probably from lack of demand.