The step from Spawn to Vampire is described on page 295 of the Monster Manual as:

If a true vampire allows a spawn to draw blood from its own body, the spawn transforms into a true vampire no longer under its master's control.

Interestingly enough, that suggests that any true Vampire can promote a Spawn, not just the Vampire that created the Spawn.

Yes (probably)

Rules as Written, I think you can kill the 0-hit-point Misty Vampire by doing damage to it three times before it escapes to its resting place, or by doing massive damage to it (equal to or exceeding its maximum hit points).

But there is certainly room for disagreement, as the rules regarding creatures with 0 hit points are written in the context of creatures that are unconscious at 0 hit points, which the Misty form of the Vampire is not.

Honestly, I don't think the intent of the rules is to allow a Vampire to be destroyed in this way. It seems that the intent is that you have to follow it back to its resting place, drive a stake through its heart, and then destroy its body while it's paralyzed.

Rules around having Zero Hit Points

PHB p. 197 says

When you drop to 0 hit points, you either die outright or become unconscious....

Creatures that take massive damage die outright.

Otherwise they fall unconscious but are dying -- they must make death saves to determine if they eventually stabilize or die outright.

There are three exceptions wherein a creature may be stable at 0 hit points -- still unconscious, but not needing to make death throws.

1. If the attacker decided to strike an incapacitating blow rather than a fatal one to take the creature down to 0 hit points [PHB p. 198]
2. If the creature is stabilized by another creature (by making a medicine check [PHB p. 197], using a Healer's kit [PHB p. 151], or by using the Spare the Dying cantrip)
3. If the creature stabilizes by rolling three successful death saving throws before rolling three failed ones.

Note that in all cases here, a creature with 0 hit points is unconscious.

Normally there is no way for a creature to be at 0 hit points and still be conscious and taking actions. However, the Vampire is clearly an exception, based on the text of its Misty Escape feature. It is very unclear whether any of the usual rules that apply to unconscious creatures at 0 hit points also apply to Misty Vampires at 0 hit points.

So What Rules Apply to a Misty Vampire that is Stable and Conscious at Zero Hit Points?

The description of the Vampire clearly admits the possibility of doing damage to the Misty Form by magical means.

Taking Damage at 0 Hit Points (PHB p. 197):

If you take any damage while you have 0 hit points, you suffer a death saving throw failure. If the damage equals or exceeds your hit point maximum, you suffer instant death.

And:

On your third [death saving throw] failure you die.

Note that a creature that takes damage while at 0 hit points that isn't enough to kill it outright remains at 0 hit points. 5e does not have a concept of negative hit points.

Two Ways to Kill the Zero-HP Vampire (RAW):

1) damage it three times, causing it to fail three death saving throws before it has a chance to stabilize or be stabilized by any of the means listed above

2) damage it once, for an amount that equals or exceeds its maximum hit points (perhaps a massive Paladin critical smite)

Caveats

1. If the Vampire has not used up its Legendary Resistances for the day, it can choose to succeed on its first three death saving throws, so you would have to hit it additional times to get three failed death saves that "stick".
2. It is very unclear whether any of the usual rules that apply to unconscious creatures taking damage at 0 hit points are also meant to apply to conscious Vampires in Misty form at 0 hit points (but RAW I believe there is nothing to override them).
3. It is possible that the Vampire's Misty Escape kicks back in and stabilizes the Vampire every time it gets hit, removing method 1 as a possibility (but I don't think so, see below).
4. It is possible that Vampires are meant to be immune to failed death saving throws (but I don't think so, see below).

On Misty Escape Re-stabilizing the Vampire

Clearly when the Vampire first drops to 0 hit points, the Misty Escape feature permits it to be stable and conscious where ordinarily it would be neither of those things.

If the Misty Escape feature triggers each time the Vampire takes damage while at 0 hit points, it could be argued that the feature stabilizes the Vampire each time.

However, the trigger for Misty Escape is that the Vampire drops to 0 hit points (outside its resting place). Getting hit while already at 0 hit points and remaining at 0 hit points does not constitute dropping to 0 hit points.

On Immunity to Failed Death Saving Throws

PHB. p. 198 says that Monsters can sometimes make Death Saving Throws:

Monsters and Death:

Most DMs have a monster die the instant it drops to 0 hit points, rather than having it fall unconscious and make death saving throws.

Mighty villains and special nonplayer characters are common exceptions; the DM may have them fall unconscious and follow the same rules as player characters.

Note that this is given as an option for a DM to make a creature more powerful than a normal monster that just drops dead at 0 hit points. But we already know that a Vampire is not a normal monster that just drops dead at 0 hit points. Its Misty Escape feature is a stronger feature than the standard Death Saves feature that prevents PCs from just dropping dead at 0 hit points. So this clause in the PHB doesn't really apply in this case.

However, I don't see anything here to suggest, whether the DM usually gives monsters death saves or not, that any monster should be immune to dying after suffering three failed death saves at 0 hit points.