MTG Salvation

What happens if for example I use Tezzeret, Cruel Machinist ability to put Azor's Gateway into play "face-down"?

Does it take on the effects of the card texts?

Would it be some hybrid "legendary land creature" that can tap for mana?

It would be a 5/5 artifact creature, with no name or other characteristics. It is NOT a land, and it can not tap for mana.
While double-faced cards that are already on the battlefield can't be turned faced down, they can be put onto the battlefield face down, such as with this Tezzeret's ultimate or the Manifest ability from Fate Reforged.

Hope that helps.

Oh well.

I have another question, if a face-down card is blinked out, i.e. exiled and then returns to play, does it retain it's face-down state?

No. By default, a permanent enters face up, untapped, phased-in, and unflipped. (Note: flipped/unflipped does not refer to being transformed, but rather to flip cards, eg. Nezumi Graverobber.)

Also note, that a card is exiled face up by default, too, so everyone sees what the card is as soon as it is exiled.

Another question to do with face-down cards.

If I have Primordial Mist creating 2/2 creatures, and I have a card that allows creatures to come back into play, say Mikaeus, the Unhallowed or Thrilling Encore, will those cards back into play, even if they are an enchantment, artifact, etc?

Quote from darrenhabib »

Another question to do with face-down cards.

If I have Primordial Mist creating 2/2 creatures, and I have a card that allows creatures to come back into play, say Mikaeus, the Unhallowed or Thrilling Encore, will those cards back into play, even if they are an enchantment, artifact, etc?

Undying (and persist) can trigger and get to resolve regardless of what card types the permanent with it has and regardless of what card types the card it becomes has (C.R. 702.93a, 702.79a). (Note that both abilities return that card to the battlefield face up if possible, even if the former permanent was face down, since they don't specify otherwise [C.R. 110.5b, 702.93a, 702.79a].)

Thrilling Encore, on the other hand, checks what card types each card in each graveyard has when it resolves (C.R. 608.2g). It will return all creature cards in all graveyards to the battlefield, regardless of what other card types they may or may not have (C.R. 608.2h). Again, those cards will enter the battlefield face up since Thrilling Encore doesn't specify otherwise (C.R. 110.5b).

EDIT: Correctness edit after comment 7 was posted.
EDIT (Sep. 1; Jan. 3): Correctness edit.
EDIT (Jul. 9, 2019): One rule was renumbered with Core Set 2020.
EDIT (Nov. 11, 2021): Some rules were renumbered in the meantime.

Note, that manifest can also put instants and sorceries face down on the battlefield. If those face down creatures die with undying or persist, they won't be able to come back, since instants and sorceries cannot enter the batlefield face up, so they remain in the graveyard. The abilities will trigger, though, they just won't be able to do anything.