Yes to both questions - all the information is public.
From the rules here with emphasis mine.
For the Gate:
Reveal the top Divinity card from each Mythology deck.
For the Divine Theatre wonder:
Reveal all the cards from a Mythology deck of your choice.
Reveal would mean the cards are face-up on the table for both players to see. They are not hidden in any way. The word reveal is used elsewhere in the rules, always to refer to public information. The setup rules make a clear distinction between face-down Mythology tokens which are then revealed. When you take a Mythology token, the rules say you "draw" two Divinity cards; which is distinct from revealing them. You secretly choose a Divinity card, it's placed face-down, and then revealed in Age II Setup.
For question 2 with the Divine Theater, while the rules aren't entirely explicit about it, it seems reasonable that since the cards are revealed, the order you stack them in is also public information. Technically this point is moot, because the Divine Theater reveals all the cards, then stacks the unused cards, but doesn't turn them face-down again. Thus both players can see the order after anyway. If this was done before activating the corresponding Mythology token, that will put the chosen and remaining cards face-down on the Pantheon and Mythology decks, so until Age II starts the other player won't know which of the two was placed.
With Enki specifically, the rules say:
When Enki is revealed, randomly draw 2 Progress tokens from those discarded at the beginning of the game. These tokens are placed face-up on Enki's card.
This means that while you're making the choice, you and the opponent both know which Progress tokens are available. The FAQ in the rules also says:
Q: What happens if Enki isn’t chosen when the Gate or the Divine Theater are activated?
A: The revealed Progress tokens are returned to the box with those discarded at the beginning of the game.
This could mean in theory that Enki is revealed twice during a game - first with the Divine Theater during Age I, not chosen, and then by being added to the Pantheon normally (or with both the Divine Theater and the Gate in either order). The Progress tokens are still returned when the Theater/Gate player doesn't choose Enki, and new ones will be drawn the second time.
