You should be fine with the battery in while playing + charging. It might charge a little slower due to the fact of your laptop is drawing more power for it's CPU and GPU, besides that, all good.
Most laptop power supplies are more then able to charge the battery and game at the same time. That said, heat is not great for the battery, so for the battery it self, maybe charging while cool is better.
I have always just left the battery in to be honest.
You should be fine with the battery in while playing + charging. It might charge a little slower due to the fact of your laptop is drawing more power for it's CPU and GPU, besides that, all good.