These are excellent questions!
So the Most Devout are a council (or conclave) of the highest ranking Septons. How they are chosen is unclear – so it could be the High Septon appointing, or it could be a more presbyterian system.
Now, historically, the Most Devout chose the High Septon from among their ranks, although that’s not a necessity. After Jaehaerys’ peace with the Church, the Crown claimed the right to name the High Septon, although this seems to have been carried out by the King nominating a candidate who would always be elected no matter what.
I find the pre-Targaryen period a fascinating mystery – I can’t imagine the rulers of Dorne or the Stormlands or the Westerlands or the Riverlands being particularly happy with the idea that the head of the Faith is so intimitately connected with House Hightower, even if that doesn’t always mean connected with House Gardener. So I would imagine there would have to be a fairly Galician system, and some very Renaissance lobbying during the choosing of new High Septons.