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.

No

There are no hidden rules.¹ There is also no general rule that effects end if their source disappears, dies, or is destroyed or anything of the sort.² The effect specifies the conditions under which it ends, and in the absence of other rules that'd add more conditions under which it would end, those are the only ones that cause it to end. The effect itself doesn't say it ends when the succubus dies, so it doesn't.

1. Can't find the reference for this, but this is a general principle in 5e
2. It's hard to prove a negative, but this is the case