1. No. Adora takes command and leads rebel attacks as herself. Sometimes completing missions without transforming. More often then not the situation gets too intense and she has to change but she simply slips away in the confusion of battle without anyone noticing. It's usually only later someone asks "where's Adora" to which She-Ra simply answers "She's safe" and no one questions it further. (leading some fans to speculate there's some kind of magic surrounding Grayskull's power that keeps them from thinking about it too much)
also it's not accurate that only three know Adora's secret. Everyone who knows Adam's secret knows Adora's and then more. There's: Adam/He-Man, Cringer/Battle Cat, Man-At-Arms, Sorceress, Orko,Spirit/Swift Wind, Light Hope, Madame Razz, Broom, Kowl, Loo-Kee, and presumably Zodac. In one episode Granamyr reads She-Ra's mind so he may know but it's not addressed.
2. The Horde rules most of Etheria the Rebellion has slowly been freeing more and more territory. The kingdom of Bright Moon, ruled by Angella, is a completely free area and there are others but the Horde still is in control of most of the planet.
3. They need a dimensional portal to travel back and forth. He-Man and company just get the Sorceress to make one for them. The bad guys are somewhat inconsistent. In one episode Skeletor needs Modulok's invention to get to Etheria and had to use He-Man's portal to get home but other times he seems perfectly capable of making a portal himself.