Turning a permanent face up or face down doesn't change whether that permanent is tapped or untapped.

If Omarthis, Ghostfire Initiate's last ability causes its controller to manifest multiple cards, those cards are manifested one at a time.

The face-down permanent is a 2/2 creature with no name, mana cost, creature types, or abilities. It's colorless and has a mana value of 0. Other effects that apply to the permanent can still grant or change any of these characteristics.

If a manifested creature would have morph if it were face up, you may also turn it face up by paying its morph cost.

Unlike a face-down creature that was cast using the morph ability, a manifested creature may still be turned face up after it loses its abilities if it's a creature card.

You can look at a face-down permanent you control any time. You can't look at face-down permanents you don't control unless an effect allows you to or instructs you to.

 If a double-faced card is manifested, it will be put onto the battlefield face down. While face down, a transforming double-faced card can't transform or convert. If the front face of a face down double-faced card is a creature card, you can turn it face up by paying its mana cost. If you do, its front face will be up.

If a face-down permanent you control leaves the battlefield, you must reveal it. You must also reveal all face-down spells and permanents you control if you leave the game or if the game ends.

If something tries to turn a face-down instant or sorcery card on the battlefield face up, reveal that card to show all players it's an instant or sorcery card. The permanent remains on the battlefield face down. Abilities that trigger when a permanent turns face up won't trigger, because even though you revealed the card, it never turned face up.

A permanent that turns face up or face down changes characteristics but is otherwise the same permanent. Spells and abilities that were targeting that permanent, as well as Auras and Equipment that were attached to the permanent, aren't affected.

Any time you have priority, you may turn a manifested creature face up by revealing that it's a creature card (ignoring any copy effects or type-changing effects that might be applying to it) and paying its mana cost. This is a special action. It doesn't use the stack and can't be responded to.

Because face-down creatures don't have names, they can't have the same name as any other creature, even another face-down creature.

Because the permanent is on the battlefield both before and after it's turned face up, turning a permanent face up doesn't cause any enters-the-battlefield abilities to trigger.

You must ensure that your face-down spells and permanents can easily be differentiated from each other. You're not allowed to mix up the cards that represent them on the battlefield to confuse other players. The order they entered the battlefield should remain clear. Common methods for indicating this include using markers or dice, or simply placing them in order on the battlefield. You must also track how each became face down (manifested, cast face down using the morph ability, and so on).

Omarthis, Ghostfire Initiate's second ability will trigger any time you put one or more counters on another colorless creature. This might be due to a spell or ability resolving or a colorless creature you control entering the battlefield with counters.