The maximum damage healable with First Aid skill is equal to the skill rating, but i am asking if
something increase possible damage healed with First Aid skill?
a. Specialization (First Aid (Combat Wounds))
b. Quick Healer Quality (+2 all Healing Tests..)
c. Something else

Nope, a and b only give bonus dice, they do not increase the skill rating, thus they do not increase the maximum number of healed boxes. So with a skill of six you could only ever heal six boxes but you would need eight hits.

The only things I can think of that would increase the maximum is the adept power improved ability, and aptitude if you actually get the skill to 7.

I use 3rd edition primarily, but have applied the damage box healing per success of 4th. I limit the max healable damage to the character's skill rating in First Aid or the Rating of the Medkit whichever is higher (medkits used to be Rating 3 on average, but I allow up to Rating 10 medical tools - but then you're tlking like a full on trauma treatment facility there). this was to reflect a good paramedic being able to stick you back together with staples and superglue or a office first aider to follow the high tech AutoDoc program of a Saviour medical pack.

With 4th edition, I've been toying with the idea of a max skill rating of 10 rather than 6 (11 if you've got the Aptitude Quality). However, to evalute between the 3rd & 4th edition, you could only reduce your 3rd ED Wound Level by 1 step with First Aid which was the equivalent of 3-6 boxes from Serious to Moderate. If you got a Deadly Wound (10 Boxes) you could only be stabilised until proper medical treatment arrived.

Remember, that quality has to be on the target, not the medic.

..A Quick Healer recovers from damage more quickly than
other characters. The character receives +2 dice pool modifier to
all Healing Tests made on/for/by him, including magical healing.

Yup, but those two dice do not increase the maximum of healed boxes. The healer is only more likely to reach the maximum.

I think the by him is referring to natural healing tests. not to there skill with a medkit on others.

Or using a medkit on themselves.

But on/for refers to First Aid or Healing Spells.

Also correct.