When we are fighting as Trogs, and playing as them our tactic cards, card such as "regroup" has no effect, right? And on the other hand "air support" adds trog unit to chosen battle objective?
I'd need to know the exact text of these two cards to answer properly.Edit 1
From each Region adjacent to the Battle Region, you may remove one of your miniatures(icon) and place it on any Battle Objective.
In a 2 or 3 player game this won't happen as Trog miniatures don't stay on the board except when locked in battle regions, so you are correct that you cannot use Regroup for the Trogs.Edit 2
Air Support: Battle:
Add one miniature(icon) from your Reserve onto any Battle Objective.
In a 2 to 3 player game the Trogs don't have a player controlled reserve, and since the card says "your" reserve I would rule that you would not trigger Air Support when playing for the Trogs in a battle. Though I would not be surprised if the official ruling allowed it to trigger in that case.
And do I place a trog nest token if I loose the fight to trogs but was still able to kill and capture all of their miniatures?
Yes, if the Trogs win control the token is placed in the region, if the Trogs lose control and have to retreat the token is placed in an adjacent region.
Another one, so for example I have two adjacent battles going on, and let say that I won in the first region, so then when playing second battle I can use "regroup" tactic card to summon one of my survived miniatures from the ended battle to the present one?
Yes this is legal, because once a Battle is over it is no longer a Battle region. Allowing you to target it for card affects.
And one more thing... When human claims a terrain control through airfield, does it activate an exploration counters? Or you have to go there with your miniatures to activate it?
It does not, only miniatures trigger tokens.