Dogmantra

The FAQ is not relevant to the Auric Runefather's ability.
The FAQ specifically refers to rolls to negate a wound/mortal wound. Negate is used as a specific game term, if the word doesn't appear in the ability, then it is not a roll to negate a wound. The Auric Runefather's ability does not negate wounds or mortal wounds, it assigns them to a different unit. The rule explicitly says that the retinue can make a ward roll for damage assigned in this way.

There's a lot of example stuff in the FAQ but the ruling essentially boils down to the one sentence in the middle:
The key words here are per attacking unit, so each time it's someone's turn to choose a unit to fight, the limit resets.
Based on your examples:
Vulkites have been chosen to fight some Zombies.
The Vulkites kill some Zombies which bounce MWs back and kill 5 Vulkites in return.
The 5 dead Vulkites can now fight on death, and they kill some more Zombies.
Another Vulkite dies to the MW bounceback. However because the Vulkites are still the unit chosen to attack and they have already fought on death, this Vulkite doesn't fight.
Later in the fight phase, the Zombies hit back at the Vulkites and kill two.
Because the attacking unit is now the Zombies, the Vulkites can fight on death again, and the two dead ones kill a couple of zombies. If any MWs bounce back and kill more Vulkites, those ones cannot fight on death.
or example 2:
Vulkites are in combat with Unit A and Unit B.
Unit A attacks, killing 10 Vulkites, these Vulkites can now fight on death.
Later in the fight phase, Unit B attacks the Vulkites and kills 20 of them.
Since it's a new attacking unit, the Vulkites can fight on death again.

Correct, the unit did not finish a pile in move within 3" of the model, so it doesn't suffer the negatives. Note that it's if a "unit" finishes a pile in, so if even a single model needs to move then the unit has made a pile in and will take the penalty.
It doesn't make it useless, but it does make it more niche, it means that enemies will have to roll higher charges and hit you front-on to maximise contact without piling in.

No, although the Cities of Sigmar units you have brought gain the Stormkeep keyword, they don't gain the Stormcast Eternals keyword. To be healed by Healing Storm, a unit needs the Stormcast Eternals keyword.
Likewise, allied Cities of Sigmar units can't be healed with Healing Storm.
However if you really wanted your Lord Relictor to be able to heal allied/coalition Cities of Sigmar units, you could give him the Heal universal prayer which can target anything.

It's not a command ability. Command abilities explicitly say that they are command abilities.
Lord of Geists is one of those slightly frustrating abilities that does two unrelated things. You don't need to issue all out attack with him at all to benefit from the second paragraph.

Au contraire! As per 18.1.1, you do have control of your home objective. Before the first turn, you automatically gain control of all objectives you are sitting on:
 

That's because the core rule is indeed a roll off for priority on turn one. This applies to Narrative and Open play.
However, the General's Handbook overrules this to mean that whoever finishes deploying first has first turn priority in Matched Play. See page 11 of the General's Handbook 2021:
 

Stormdrake Guard cannot retreat with their hero phase move. Their hero phase move allows them to make a normal move, and a retreat is not a normal move (See 8.0 Movement Phase in the core rules).
Even if they could, 8.2 Retreating states that a model that retreats "cannot shoot or charge later in the turn"

Rule 27.3.1 Enhancement Restrictions (pg 90 in the GHB) states
So unless the artefact description specifically says you can add it to mount attacks, no you only add to the actual unit's attacks.

You have to pick one. I am away from my book at the moment but the term to look up in the core rules is "triggered effects" and it essentiallu says that each die roll can only trigger a single effect like double hits or mws.