I don't believe that your H is gay. If he lived in an earlier era, he would be what was referred to as a "dandy". I think that he is heterosexual but just more "top" than "alpha". IMO, the best way to figure out how Alpha a guy is to observe how he interacts with children. You have indicated that your H actually "fights' with your kids on their level. Very bad sign in my opinion. I think the more Alpha a guy is the more he will give off a protective, responsible or even semi-maternal vibe around children, although both kinds of guys can playfully interact with kids at times.
OTOH, I think Karen's husband is Alpha but also gay. Our society just doesn't do a very good job of recognizing that men can be both Alpha and gay.
"Tell me, what is it you plan to do with your one wild and precious life?" - Mary Oliver