Not knowing if this is related to engine overheating, hence potential replacement of wp and t-stat, here is what I need to report:

First start in the morning, engine idle in P, no AC: the fan would come turning in very high speed in just a couple of seconds, with very loud fan noise that can wakes up everyone in the neighborhood and no stop. All this without any overheating warning on the dashboard - the overheating warning would come up in a couple of minutes once the vehicle is on local street, then it goes in to a limp mode, disabling any driving capability.

What puzzles me is this: while engine started cold in the morning, why the fan got turning so quick and at such high speed?