Hmm max VG...Then I would not think it would be what little Pg there is in it causing the cough. It could be one of the flavors or an additive like EM. I know that if there is a lot of peach flavor in a juice I cough uncontrollably. Not sure why and I hate it because I love peach.
Now the battery. Split the coating - I am guessing you are talking about the plastic (or whatever it is made of) wrapper. Honestly I am not sure. If it was me I would wrap it with tape or something. I look at it like I would rather be safe than sorry. The body of a battery is metal so I guess it could potentially cause a short. This is if the entire wrapper was removed. Like I said I am not 100% sure.
My main concern is the battery getting hot. If I remember correctly you are vaping 1 ohm and higher. If that is true and your battery was getting really hot then something was wrong. I chain vape on 0.6 ohm and my battery has never even got warm to the touch. Check the ohms of your rda and make sure the reading is constant and not bouncing all over the place. Also while it is on ohm tester lightly touch your coils and watch to see if the reading starts to bounce. Sometimes a leg will break and the end will still be touching the post but not making a 100% connection. Another thing is to make sure your top cap is not touching your coils and also that your coils are not touching the build deck.