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.
Unless the description states the target is aware of the spell, it is not automatically aware by default.
Spells only say what they do. If a target knew it was under the effects of a spell any time a spell was cast on it, then illusion spells would be wholly useless.
Since your question is about charms/enchantments, it's worth pointing out that these spells explicitly state when a target is aware. For example, Friends states the target knows it was magically compelled. Charm Person also states this.
An example of a spell that doesn't state this is Geas. At no point does it say that the affected creature is aware that it is under magical compulsion to obey the command. This is important because if you happen to be a Sorcerer, you could cast this as a subtle spell and the affected target would have absolutely no way to tell it had even been targeted by a magical effect.
By and large, the default for all magic effects is that something is unaware until they discover it through some mechanism. For instance, you would automatically see a fireball coming at you because it's not a hidden object. Whereas you may not notice a delayed blast fireball that's hidden above a door jamb as you passed under it. Other methods of discovering a spell is affecting you are things like, "Hey, the whole world is getting a lot bigger! Hmmmm.... I might be getting shrunk by an enlarge/reduce spell." Or, "How come I have hooves and oh my god I'm a goat." when targeted by a polymorph.
At the end of the day, the question you need to ask is, "Does the spell explicitly state that the creature is aware?"
Worthwhile contribution from Rubiksmoose:
"You and your companions might deduce that you were beguiled if evidence of the spell is found. It’s ultimately up to the DM whether you discover the presence of inconspicuous spells. Discovery usually comes through the use of skills like Arcana, Investigation, Insight, and Perception or through spells like detect magic." http://rpg.stackexchange.com/a/92279/28591
Best Answer
You cannot intentionally lower weapon damage, but you have a weaker unarmed strike that deals less damage.
The only (officially published) fiend with the Fiendish Charm feature is the Cambion. The relevant portion reads:
So the idea here is that our ally is charmed, and we want to do some damage to them (harm) in order to give them an opportunity to retry the saving throw. You cannot intentionally lower the damage of the attack, there is just no rule that allows that. The rules for Making an Attack state:
And that's it. No ability to "pull your punches", so to speak, is given. However, punches may be a viable solution here. If you are not playing a monk that assigns a damage die to unarmed strikes, the following rule applies to all unarmed strikes (and optionally if you are a monk):
This effectively minimizes the damage you do with a melee attack, which is essentially what you're looking for here. Instead of slapping your ally with your sword, give them a good kick in the pants for less damage.