A partial answer to the question about the Gleason cover of the unit interval: it can be found everywhere in every compact extremally disconnected space.
The point is: in a compact extremally disconnected space the closure of every countably infinite relatively discrete subset is (homeomorphic to) $\beta\mathbb{N}$ and $\beta\mathbb{N}$ contains homeomorphic copies of all compact extremally disconnected spaces of weight $\mathfrak{c}$ (or less), see [this paper for example][1].


  [1]: https://zbmath.org/0615.54004