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.
The Dominate Beast spell will end
There was a (somewhat confusing) developer tweet about this topic. Jeremy Crawford stated (emphasis mine):
In #DnD, the exceptional trumps the general. (No longer being a valid target trumps condition carryover.)
When a wildshaped Druid reverts to their normal form, we have those two conditions: the target changing type, and condition (charmed) carrying over. The target changing type makes it an invalid target which, as confirmed in the sage advice tweet, trumps the charmed condition carrying over.
Since the spell would no longer has a valid target once the druid changes back, the spell will end.
What about the other two Dominate X spells?
For completeness, let's run through the two other dominate spells.
Dominate Person requires a humanoid target. All official PC races are humanoid, so while the druid is in its normal form it can be targeted and charmed by a Dominate Person spell. If the druid wildshapes into a beast, its type changes to match that beast. So, the druid is no longer a humanoid, the target of Dominate Person is invalid, and the spell ends. In this way, it works in reverse of the Dominate Beast spell.
Dominate Monster works on any creature. Whether in beast form, humanoid form, or even in a Circle of the Moon druid's elemental form, the druid is still a creature. Thus, even going in and out of wild shape won't make the druid an invalid target. Since the target is still valid, the Dominate Monster spell will still have full effect, even if the druid wild shapes.
Best Answer
Mind control is almost always left up to the GM
Technically, you simply cannot tell a creature to "use a legendary action" as those don't exist in-character; you'd instead have to describe what you want the creature to do and then the GM would determine how that plays out. This is the case for mind control in general, and this is no exception.
When you tell a monster to do something, perhaps the easiest way for them to do that would be by using their legendary, lair, or other sorts of actions. Whether the creature actually uses these is the GM's decision because the game's rules are silent here. There's no exacting description or definition of the extent of what you can force a creature to do through generic commands.
That said, let's look at dominate monster more closely:
You cannot specifically tell a monster to use a particular legendary, lair, or other sort of action because that command would be far more specific than a general course of action. In order to do this, you would have to take total and precise control of the target, which the spell explicitly allows you to do. Then you would have to describe what you want the creature to do and the GM would narrate the results.
However, again, if the generic course of action is best carried out through specific actions it will fall to the GM to determine what actually happens at the table. This is the case for almost every single command you could give - the GM determines what happens.
Similarly, the Necromancy Wizard's Command Undead feature says that the creature "obeys your commands". What this means, what you can command it to do, the amount of knowledge you have about its available actions, and the precision allowed in your commands is entirely going to be up to the GM.