The answer is
the Puzzling.SE moderators: Deusovi, Emrakul, and GentlePurpleRain (in that order).
They live in the eminence, the mountains high
Createth the doctrines, and abides by.
O Potentates, O Overlords, t'is not a wry.
Incarcerate, excutes - nefarious they sly.
They can be seen as being set 'above' regular users, although they don't actually create the site 'doctrines' or policies, but only enforce them.
The amnesiac phantom is already dead
Gods of birds, maker of dread !
Brother of conspicuous, impish he shed.
This is DEUSOVI: he doesn't sleep much, and we sometimes joke about him having a sock ("brother") called Obusovi. "Impish" ~ devious - an anagram of his username.
The titan Eldrazi, is dread as a whole
Egyptians sun deity too, not very ole.
Sounds frosty in the end, uplifter of soul.
Apparently EMRAKUL is "the largest and most fearsome of the Eldrazi". The Egyptian sun god is RA, and the final line refers to KUL or "cool".
Placidly peaceful, pleasant person.
'en Pomegranate periwenkle plum pigmentation
Precipitation, pouring. (Not a pun !)
GENTLEPURPLERAIN's profile is full of P-alliteration. Also, the first line means "gentle", the second "purple", and the third "rain" (thanks @F1Krazy).