According to the response from David_K, the answer to my question is:
Yes, the Nest batteries will charge from the line connecting the thermostat to the furnace. If there is a circuit fault, the thermostat can stop charging. That happened in our case with Error E195.
I couldn't find a bad (browned, loose, disconnected or otherwise) wire on our Carrier furnace when searching for a problem wire. Had to call the service company. They found a bad wire but had to really search and search for it. It was browned as though the furnace had really been working hard. It took 4 hours to locate. Surprisingly, the suspect wire was not near the furnace flame jets or anywhere really near what might be an obvious heat source. It was way around the backside of the furnace. We had an extended period of weather where the cold approached -40C/-40F about a month ago. The furnace was under load and definitely stressed during that time. The initial tech called in the boss to help after 2 hours.
We maintain our filters and have been changing them every 3 months. Furnace tech suggest we do not buy the expensive super filters that are supposed to maximize air quality. He said buy the cheap ones and replace every 2 months. Let the furnace breath. He also left the filter sticking out of its filter slot by about a half an inch to ensure the furnace can suck enough air under heavy load. That of course also means leaving the sheet metal filter cover off. That's the cover that is the same approximate width as the thickness of the filter.
Following his recommendations. Interestingly, we discovered a single fuse on the control board of the furnace had been replaced somewhere along the line and not by us. The fuse was a 20 amp fuse in a 3 amp slot. The house came with the furnace and we were simply unaware of that situation. I asked if that might have been the cause of the problem. He couldn't really answer definitively but said yes, it could be. I have my suspicions though.
