A full scan will reveal any anomolies in the coolant temp sensor and other
inputs that could trigger this isssue.

I don't know, but I am guessing that the computer may turn the light on if
it takes too long to get to operating temperature.

A properly operating thermostat will allow the engine to get to operating
temperature regardless of outside temperature. I'd replace the thermostat.

At least one fan will come on if the A/C compressor is running. Both will be
on if the computer is in "Limp Home Mode". The Check Engine light should be
on if this is the case.

It is possible that you have two problems: the thermostat and a temperature
sensor. There are two temperature sensors, one that the gage uses and one
that the computer uses. Look for loose wires before replacing parts.

Also check coolant level and repeat the bleeding procedure.