1. There are some cards that allow you to take a card of your choosing from the shop without paying its cost. When you are presented with this opportunity do you add a fifth card to the shop or do you just choose between the four cards in the shop?
I'd say that only happens with an actual Shop action, so no.
2. If a card allows you to take a card from the shop without paying its cost, does that mean you are also allowed to recruit a companion even if you dont have the required SPECIAL token?
Yes. That is its cost.
3. If you get a companion through the loot deck (has no SPECIAL token requirement) that need you to be not "vilainified" when you unexhaust or it will be discarded, can you still equip it even if you are "villanified" ? There seems to be no requirement to not have the villanified trait to equip it, just to keep it after unexhausting it.
Yes. You can equip it, just understand it might not last very long.
4. When I take a camp action, can I choose not to unexhaust a companion? And do unequiped companions unexhaust when camping or only equiped companions?
Un-exhausting is a mandatory action, and affects both equipped and unequipped cards.
5. Are there only 4 progress stages in the faction progress bar?
The colors of the bars indicate the stage right?
It seems like few stages considering there seem no way of pushing back a factions progress.
There are 7 stages -- 3 groups of 2, plus a last one which indicates end-of game. The colours are there to indicate different values of the X and Y variables, but otherwise the two spaces within one colour are separate.
Is there someway to push back the progress of a faction?
I've never seen a way, but I haven't played all the scenarios. I suspect not.