>How was/is this religion spread
The Satyr upper cult infiltrated faun societies some hundred years ago in a memorable way, and I assume that that counts. Generally, one satyr and one satyr-loyal faun "bait" moved into a nearby cave, or village, respectively. The Faun bait spun amazing tales of the power to make others jealous of your amazing charisma and silken tounge, and took young, promising Faun to see the preacher, where he preached to the few faun who wanted to listen, turning them from simple, life-loving faun, to cruel, opportunistic Satyr. Then, the preacher moved on, and the satyr moved to a satyr village hidden deep in the wilderness away from prying eyes, and built their strength in a new society, a society shaped by the domination of the strong, and the fluid hierarchy of pure force of character, and physical strength.
>What are the tenets of this religion
Already said, oops. Anyway, strength, the particular lack of respect for arbitrarily decided leaderships, and domination. Also, lust and virility are big ones.
>What is the hierarchy of this religion
At the top are the re-incarnation of the gods themselves. Of course, they are not respected, and tend to get assassinated. Very quickly. Then, below those, are preachers, those who are able to spread the gift of Satyrism. Under those are the Preachers' apprentices, who work as an entourage, harem, and underlings in general to the preachers. Under those are the common people. Of course, the church is separated from the state, and indeed, killing a preacher is no more frowned upon than killing an urchin.
>Is there an adversary figure or place of punishment for those who went against the tenets of the religion, or both?
Naye. The tenets are being strong and dominant however. So, if you fail that, you have an extremely bad life in front of you, as a doormat, victim, and what not. You are practically worthless.