as i remember, after a battle the surviving troops that been damaged but not enough to die will have lower hp than their original life before the fight. so will their hp "regen" after a battle or will it remain the same?

Units won't necessarily have low HP after a fight. Some might, some might not. You can check unit's HP from the orders menu and clicking on the squad and then the eye icon. HP will only regen if your squad is docked (not moving) for one or more ticks. The amount of HP regenerated will be the amount of ticks passed.

If there's radiation on the location, then it will not regen until you leave the place or if the radiation goes away. Radiation will take away 1HP per tick, but will not kill - meaning lowest HP it can bring your units to is 1HP.