#1 does the fan ever stop running?
The unit uses variable speed indoor and outdoor fans. Just as a reference so you will understand what’s going on, as the load increases or when you lower the cool setting or raise it for heat, the compressor also ramps up speed variably to meet the load. Simultaneously, the outdoor fan speeds up to meet the higher discharge pressures of the compressor and if the indoor unit is set to auto, it will also speed up.
So most likely unless you have the unit OFF, the fan will be turning but if the temperature setting and the room temperature are very close, the speed of the fan will run very very slow until the load increases again. This is actually a good thing as it will help the system maintain a constant movement of air to keep the spaces humidity low and no drafts like you would get from a standard window unit.