Question 1:
Is Brew Elixir an action point?
Is Drinking Elixir an action Point?
Is Inky Substance an action?
Is Protective Tonic?
etc.

I'm asking more or less about all skills for Apothecary.
Does every card that does not have the action turn(The curved arrow) cost an action as well?

According to the Hero Turn Summary it says
"3. perform 2 actions.
Attack, search, Move, Special, Rest, Stand Up, Perform an (curved Arrow) Or Skill, Open/Close Door, Revive a Hero."

According to the players I'm playing with, they read the rules as such that Brew Elixir cost a turn and that there exist no "Free action" in this game.

What is correct?

If a skill does not say "as an action" or have the "action arrow," it does NOT cost an action. It may still cost fatigue (bottom of the card). Furthermore, these abilities can be used more than once per turn as long as the card doesn't need to be exhausted (or otherwise say "limit once per turn" or "once per round").