Yes to all.
Is the creature still under the spell's effect after being reduced to 0, then brought above it?
The spell has not ended, and you haven't chosen a new target. This can be useful if, for example, your target was merely unconscious and you wanted to squeeze as much damage as possible to kill him, since, when at 0HP,
If the damage equals or exceeds your hit point maximum, you suffer instant death.
While the creature is again above 0 hit points, can the caster use their bonus action to designate a new target for the spell?
RaW, yes. The spell's description wouldn't take into account a target possibly being healed back up. A spell like Disintegrate was eventually errata'd to handle such scenarios (in the cases of Druids, specifically), so maybe the intent is that you could move the Hunter's Mark only if your target was unconscious/dead. However, as it is currently worded, even if the creature is brought up from 0HP, it met the condition on Hunter's Mark and Hex.
If the target drops to 0 hit points before this spell ends,
So you can change the spell's target.
Does choosing a new target end the spell on the first one?
Yes. It seems only 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's the spell's intention, and designers use the terminology "move" without issues when referring to targeting new targets.
That being said, the wording is ambiguous.
You place a curse on a creature. [...] You can use a bonus action [...] to curse a new creature.
Does this imply the old one is ignored? If we read the entire spell's description, it only makes sense so.
Until the spell ends, you deal an extra 1d6 necrotic damage to the target when you hit it with an attack. Also, choose one ability when you cast the spell. The target has disadvantage on ability checks made with the chosen ability. If the target drops to 0 hit points before this spell ends, you can use a bonus action on a turn of yours to curse a new creature. A Remove Curse cast on the target ends this spell early.
The spell always refers to a single target, not to cursed targets. Also, if you had multiple targets, by this wording, casting Remove Curse on one of the targets would dispel all of them entirely. Your DM can make some house-rule based on this ambiguity, yes, but I feel all indications point towards the spell being meant for single-target, ever. Anyhow, I've asked on Twitter, and maybe we'll have confirmation.