Hi 1stTimeLaptoper, welcome to the HP Forums. First of all, let me say that this was an excellent post! You are right to be concerned with the temperatures. The lifespan of a processor is directly related to how hot it gets and how often it gets hot. The troubleshooting you provided was right on the money for the first step. Did you also follow the link at the bottom for: Fan is Noisy and Spins Constantly, PC is Warmer than Normal
It is a journey to get to the Fan in a notebook. I assume you are not within the warranty period, otherwise you would have called for warranty repair right?
I have included the: HP Pavilion g6 Notebook PC Maintenance and Service Guide. In this, page 69 leads to the instructions to get to the Fan & Heat Sink.
If you do go so far as to reapplying thermal paste, I recommend reviewing this older YouTube video, for the application process: Installing Thermal Compound (NCIX Tech Tips #11)
I hope this helps.