Can the zombie be charmed into fighting me?

No.

In the specific case of the Succubus (and the spell Charm Person which the Charm action seems to borrow from) the target has to be a humanoid. A Zombie is of the type undead, and thus not a valid target. Though there are other spells and abilities that produce the charmed condition on any type of creature who is not immune to the charm condition, like Fey Presence and Hypnotic Pattern.

Moreover, the charm effect doesn't grant control, it would only prevents it from attacking the charmer, and gain advantage on social checks.

A charmed creature can't attack the charmer or target the charmer with harmful abilities or magical effects.

The charmer has advantage on any ability check to interact socially with the creature. (PHB 290)

However, since the undead is compelled to follow the command of the person who controls it, the Zombie's new found love for you shouldn't prevent it from carrying out the task.

In general, can creatures you command (familiars, summoned, animated) be charmed into fighting its master?

Charmed

Charmed doesn't allow control. Familiars would be susceptible to the charmed condition unless their stat block provides immunity.

Suggestion

Suggestion won't work on Zombies. Suggestion reads:

.. influence a creature you ca see within range that can hear and understand you. Creatures that can't be charmed are immune to this effect.

Zombies don't have drive/will, so even if someone suggested to the Zombie should attack you, it would be overridden by your command, as long as you are in control. Even outside of a caster's control, a Zombie doesn't care about anything except killing the nearest target; so suggesting outside of who or what to attack next might not have the intended affect -- but it up the DM what the Zombie does with the suggestion.

A Find Familiar (both typical and Pact of Chain) summoned familiar would arguably be susceptible to suggestion; and the variant familiar Imp, Quaist or Pseudodragon would definitely be susceptible to suggestion.

Dominate

Dominate person wouldn't work on a Zombie, as a Zombie is not of type humanoid, it is of type undead. Dominate Monster (lvl 8 spell) would work to control a zombie.

Dominate Beast on a Find Familiar summoned familiar probably shouldn't work, because it isn't a beast, it is a spirit -- but Dominate Monster should, as it says specifically "creature" which includes spirits.

The variant familiar Imp, Quaist or Psudodragon would be a real creature, but all count as monsters by their type, so Dominate Monster would apply (not beast).

Summoned Creatures and constructs

Summoned creatures and constructs are all called forth from their own spells with their own verbiage. Some can be freed much more easily from the hold of the summoner than others. Most of them you lose control of by losing concentration. In either case, there are typically ways of using them against the caster.

Charms don’t override/preempt each other unless that is called out in the rules

The quick answer is no — abilities or spells that impose the charmed condition don’t cancel previous charmed conditions. They can be in effect at the same time.

The answers to this question covered some of these charming rules

One exception is in the case of harpies, once a target is charmed by a given harpy, other harpies cannot charm the creature.

Charms are not all equal

Things get complicated because there is not a single “charm effect.” Various spells or abilities which impose the charmed condition (many of which are called “charm” or include the word in their name) actually impose very different sets of effects (See below).

For example, succubus can command a humanoid affected by her charm ability, while charm person only makes the target regard the caster as a “friendly acquaintance.” In general, the succubus’ commands would take priority over the request made with the benefit of Charm Person.

Charms in direct opposition

The exception to the above is if the command directly contradicts an effect of the other power, or contradicts the charmed condition, for example, if the succubus orders its thrall to attack a wizard who charmed the same humanoid. The charmed condition precludes this (see Appendix A: Conditions, PH, p.290).

In that case, the DM may rule that a contest is in order (See Contests, DMG, p 238) or may rule that the charmed condition precludes the action.

Crown of Madness vs. Vampire Charm

In the question’s particular case, the caster of Crown of Madness could maintain control the charmed character as long as the caster meets the requirements of the spell. That is, choosing a creature for the charmed PC to attack before moving on the first round (which might necessitate the PC attacking an ally) and “using his action to control the target” on subsequent rounds (PH p.230).

Also note the vampire’s effect might outlast Crown of Madness. The spell would persist while the caster’s concentration lasts, up to 1 minute. After that, the vampire’s charm, which lasts 24 hours, would still be in effect.

If the caster ordered the charmed PC to attack the vampire, then that would be a Contest as described above.

Differing Charm Effects

Here are a few charm effects for reference. There are a lot of subtle differences, so you have to be aware of the specifics to determine how they would interact.

Crown of Madness (PH p.229):

The charmed target must use its action before moving on each of its turns to make a melee attack against a creature other than itself that you mentally choose

Vampire (MM p. 297):

the target isn’t under the vampire's control

Charm Person (PH p. 221)

The charmed creature regards you as a friendly acquaintence.

Dominate Person:

While the target is charmed, you have a telepathic link with it...You can use this telepathic link to issue commands...

Succubus (MM p. 285):

The charmed target obeys the fiend’s verbal or telepathic commands.

Harpy (MM p.181):

While charmed by the harpy, a target is incapacitated and ignores the songs of other harpies.