How does the Gift of the Ever-Living Ones Invocation interact with Vampiric Touch? Gift of the Ever-Living Ones maximizes healing, while Vampiric Touch deals damage while healing you. Does Gift of the Ever-Living Ones automatically make it so you deal full damage, or not?

You're not rolling to determine how many hit points you regain in Vampiric Touch, you're rolling to determine how much damage you deal. They don't work together.

Or.... would it stack with the necrotic damage of Hex.. or any other necrotic damage done simultaneously (fallen Aasimar racial ability)? 

I'm thinking this because although it says "use xd6 for damange', it then says that you regain half the necrotic damage dealt.... and since it's done simultaneously there is what to think about (because the upcast would be more then the 3d6s originally included too)

I would allow it 

Whenever you regain hit points while your familiar is within 100 feet of you, treat any dice rolled to determine the hit points you regain as having rolled their maximum value for you.

treat any dice rolled that determines the hit points means that if you take half of vampiric touch that max damage 

This is a simple overreach of the rules while trying to stretch the awesomeness of the invocation. All of your options deal damage and you heal the full amount of damage taken.

One is a roll to affect another creature and damage it.

The other maximizes what you receive out of the damage dealt.

The two are not intertwined. If their was some ability that ******** the ability to heal from any of the mentioned abilities/spells, then it would not work on you depending on how strict your DM wants to be. Certain powers supersede others like light in a darkness spell. I have this invocation on my character and it makes combat healing effective which I never thought I would ever say.