Why does the outside fan stop but the blower fan continues to run

If the blower fan assembly turns freely when when it is removed from the forced air unit then there must be either something that is adjacent to where the blower mounts that has gotten in there. When the blower is out check very carefully for anything that may be in there. Maybe there is a flap, piece of insulation or even a dead animal that got into that area.

From the description given it was not fully clear, but it sounds like the blower fan is directly mounted to the motor shaft as opposed to there being some type of drive belt assembly between the blower and the motor. It is possible that there may be an internal failure inside the motor that is causing the problem. Normally you would be able to detect that when manually turing the motor - but not always. If there is a separation between the two then you would need to check both ends of the unit - motor and blower - plus the belt.

You're looking for a specification called the operating temperature range of the thermostat or router or both. This is the temperature range within which the manufacturer states that the device will function correctly.

For consumer electronics like these, it's typically from 0°C (32°F) up to about 60°C (140°F) to 70°C (158°F). So if you turn the heat to off, the ambient temperature in the house will drop below that range, and the thermostat will stop working properly.

There should be a specifications page in the manuals for the thermostat and router that will tell you their minimum operating temperatures. Keep your house above the upper of those temperatures and you should be OK.