Is the creature still under the spell's effect after being reduced to 0, then brought on top of it?
The spell is not over and you have not chosen a new target. This can be useful if, for example, your target was simply unconscious and you wanted to inflict as much damage as possible to kill him, because at 0HP,
If the damage is equal to or greater than your maximum health, you suffer instant death.
While the creature is again above 0 health, can the caster use his bonus action to designate a new target for the spell?
RaW, yes. The description of the spell would not take into account the fact that a target could be healed. A spell like Disintegrate was finally used to deal with such scenarios (in the case of Druids in particular), so maybe you could move the Hunter's Mark only if your target was unconscious / dead. However, as currently written, even if the creature is raised from 0HP, it fulfills Hunter's Mark and Hex condition.
If the target drops to 0 hp before the spell ends,
So, you can change the target of the spell.
Does the choice of a new target end the fate of the first?
Yes. It seems logical that targeting a new creature excludes the previous one. Otherwise, you would now have a single target spell targeting 2 creatures (or more, if more were brought to 0HP). At least that is the intention of the lot, and designers use the terminology "move" without problems when they refer to targeting new targets.
That being said, the wording is ambiguous.
You place a curse on a creature. […] You can use a bonus action […] curse a new creature.
Does this imply that the old is ignored? If we read the complete description of the spell, it only makes sense if.
Until the end of the spell, you deal 1d6 additional necrotic damage to target when you hit him with an attack. Also choose an ability when you cast the spell. Target has a disadvantage on the capacity checks performed with the chosen capacity. Yes target Fall to 0 points before the end of the spell, you can use a bonus action each on your side to curse a new creature. A spell spell on the target ends that spell early.
The spell always refers to a Only target, not to cursed targets. In addition, if you had multiple targets, by this formulation, throwing Remove the curse on one of the targets would dispel them entirely. Your deputy minister can establish an internal rule based on that ambiguity, yes, but I think there is evidence that the fate is for single target, never. Anyway, I asked on Twitterand maybe we will have confirmation.