Is it OK to plug in laptop to wall outlet while it is turned on (running on battery)?


Yes it is.



All the electrical stuff for any product is in the front of the manual, namely the warnings. If the warnings doesn’t include “Don’t plug in while laptop is on,” then you’re fine.

Teaching a man to fish, here.

It’s not safe at all. In fact, you should shut the laptop down and allow it to cool completely before you connect it to the outlet. If time is tight then you should put it into a bath of cold water and I would suggest drilling holes in the screen to improve airflow in that area of the device.