The creature's "enters" abilities (and any other abilities relating to the creature entering) do not trigger when it turns face up.

Only the controller of the face down creature can look at it.

The creature enters face down, so none of its "enters" abilities will trigger or have any effect. Also none of its "As this enters" abilities apply.

The effect that turns it face-up is a replacement effect. It doesn't use the stack and can't be responded to.

Illusionary Mask's ability will continue to apply to creatures cast face down with it, even if Illusionary Mask has left the battlefield.

While the creature card is face down, it's a 2/2 creature with no name, mana cost, color, creature type, abilities, or expansion symbol. Since it has no mana cost, its mana value is 0.

You may look at a face-down spell or permanent you control at any time.

If you use the ability to cast a creature card face down, you must keep track of the amount and types of mana you spent on {X}. If that creature spell is moved from the stack to anywhere other than the battlefield, the resulting creature leaves the battlefield, or the game ends, the face-down card is revealed. If its mana cost couldn't be paid by some amount of, or all of, the mana you spent on {X}, all applicable penalties for casting a card illegally are assessed.

Both the amount and types of mana you spend on {X} are taken into account while you're choosing a creature card from your hand. For example, if you spent {U}{U} on {X}, you can choose a creature card with mana cost {U}{U}, {1}{U}, {2}, or {W/U}{W/U}, among other possibilities, but not one that costs {2}{U} or one that costs {G}.

You may not turn a face-down spell face up. You may not turn a face-down permanent face up unless it would have morph while face up or an effect specifically allows you to turn it face up. Illusionary Mask's ability has you turn a face-down creature face up if it would assign damage, deal damage, be dealt damage, or become tapped, but not for any other reason. For example, if you use Illusionary Mask's ability to cast a black creature face down, you can't turn it face up just because it's being targeted by Terror.

You actually cast the card face-down, much as when playing a spell with Morph. It can be responded to and countered.

You can turn a face-down permanent face up if it would have morph while face up. This applies to creatures you cast face down as a result of Illusionary Mask's effect. The rest of Illusionary Mask's effect applies to it as well.