Trya having a friend start the engine while you listen under the hood to try to determine where the screeching is originating from.
In addition to a loose belt making noise in cold weather, noise can also originate from the bearing sof any of the driven components, including the tensioner, the alternator, the power steering pump, and idler pulleys. I’d consider the AC as unlikely unlesss you have it on in the morning (which includes having the defroster on).
Since it’s being affected by pushing the clutch pedal in, I think the release bearing should be considered suspect as well. How old is the clutch assembly?