If the second is ticking as normally it is not the battery. If the battery is low the second hand will advance only at 5 second intervals meaning for four seconds the second does not move and on the 5th second it will advance by five seconds. This is a feature unique to Tag Heuer.
Whatever will turn the screw without stripping it. Also be careful with those little screws if you drop it on carpet it will take a miracle to find it.
It sounds like there may be debris in the gear train. If you bring it to your local watchmaker or jewelry store, they can clean it out for you. you shouldn't have any problems after you do that
Tags have a screw down crown (winder knob) turn the knob counter clock-wise until you feel the crown unscrew (not off) then pull the crown carfully. It will pop once this is for date adjust. Then pull once more and you can now adjust the hands, set accordingly then push crown back in. After you push it back in gently compress the crown and tighten clockwise until it stops. CAUTION!!!!!!! Tags are not waterproof if the crown is not screwed back down. I hope this helps .
As long as they don't charge you $40, it should be free. Having said that my machine for testing water resistance cost almost $5000. I have to recoup that somewhere. Now you know why it cost so much. Does it require testing, no. But kon't come crying to me your watch leaked and I was the last to open it. I told you it needed to be tested but you didn't want to spend the $40. That is the view from the other side of the bench.