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What happens when I reanimate a creature with Animate Dead which has protection from black? Will the creature keep being in play without the enchantment or die? Or can the creature be a legal target even in the graveyard?

You can cast Animate Dead on a creature card with protection from black because protection doesn't work in the graveyard. The creature will return to the battlefield and protection starts working again, so Animate Dead will go to the graveyard immediately. As a consequence of that, you have to sacrifice the creature, but the creature will stay on the battlefield long enough to interact with it.

Animate Dead starts out as an aura that can "Enchant creature card in a graveyard". While a card is in the graveyard, it has all its abilities including protection, but they don't work unless otherwise stated:

112.6. Abilities of an instant or sorcery spell usually function only while that object is on the stack. Abilities of all other objects usually function only while that object is on the battlefield. The exceptions are as follows: [none apply]

Therefore you can target and enchant that creature card with Animate Dead. While resolving, its effect will try to attach it to the creature, but that creature, in contrast to the creature card in the graveyard, has a functional protection from black, so it will not attach:

702.16c A permanent or player with protection can’t be enchanted by Auras that have the stated quality. Such Auras attached to the permanent or player with protection will be put into their owners’ graveyards as a state-based action. (See rule 704, “State-Based Actions.”)

701.3a [..] An Aura, Equipment, or Fortification can’t be attached to an object it couldn’t enchant, equip, or fortify, respectively.

Next, the active player gains priority:

116.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.

Before a player gains priority, state-based actions are checked:

704.3. Whenever a player would get priority (see rule 116, “Timing and Priority”), the game checks for any of the listed conditions for state-based actions, then performs all applicable state-based actions simultaneously as a single event. [..]

Animate Dead is not attached to an object, so it goes to the graveyard:

704.5m If an Aura is attached to an illegal object or player, or is not attached to an object or player, that Aura is put into its owner’s graveyard.

Therefore, before the active player gains priority, Animate Dead will leave the battlefield and its sacrifice ability triggers. It is a normal triggered ability that uses the stack, so the creature remains on the battlefield without Animate Dead attached to it until that ability resolves. Players can respond to that ability with instant-speed spells and abilities, for example by bouncing the creature or using its activated abilities.

The creature will die, see the rulings below the card:

If the creature put onto the battlefield has protection from black—or if the creature can’t legally be enchanted by Animate Dead for another reason—Animate Dead won’t be able to attach to it. It will be put into the graveyard as a state-based action, causing its delayed triggered ability to trigger. When the trigger resolves, if the creature’s still on the battlefield, its controller will sacrifice it.

So the creature does enter the battlefield, since in the graveyard it doesn't have protection from black. This might be important for other triggers like Ajani's Welcome. But it's guaranteed to leave the battlefield very soon after.

I assume you mean Animate Dead. If so the rulings on the Gatherer page state:

If the creature put onto the battlefield has protection from black—or
if the creature can’t legally be enchanted by Animate Dead for another
reason—Animate Dead won’t be able to attach to it. It will be put into
the graveyard as a state-based action, causing its delayed triggered
ability to trigger. When the trigger resolves, if the creature’s still
on the battlefield, its controller will sacrifice it.

The abilities on creature cards while on the graveyard are not relevant unless otherwise stated (same happens on your hand).
when you cast Animate Dead, you are able to target the creature with pro black in the graveyard. This creature will then be returned to the battlefield and will be enchanted by Animate Dead. At this point and before you gain priority (or your opponent) state base checks are done, the game realizes that the enchanted is attached to an illegal target and is then sent to the graveyard. This will then trigger Animate Dead's trigger of leaving the battlefield and the creature goes with it.

502.7c A permanent with protection can't be enchanted by enchantments that have the stated quality. Such enchantments enchanting the
permanent with protection will be put into their owner;s graveyards as
a state-based effect (see rule 420, "state-based effects")