In my experience, the fhe fan clutch should be pretty tight when stone cold, and immediately begin to loosen up when cranked. It should not make too much noise even when revved up 30 seconds or so after a cold start. When warm or hot, it should move quite freely, spin at least one full turn when spun by hand. Here in Texas I've never seen mine tighten up when turning off a hot engine and letting it heat soak. The ONLY time I hear it tighten up when driving is when the AC is NOT on and the engine sits at a red light for awhile, taking the temp above 100 but not to 110. Then it will rev audibly for a few seconds when starting off. When the AC is on, the low electric fan comes on via a pressure switch and keeps the temp lower, and stops the clutch fan from tightening up. With AC off the low fan is off always, so the motor heats up more standing still. Then the hi fan comes on at 110, this will declutch the clutch fan in a few seconds.
my fan has always had a little play in the clutch - still going after 20 years though.

The cold noise could be the smog pump; they only run for about 5 minutes after a cold start, and they make that exact noise. Easy to check, you can trace the wire back to the fender and unplug it for awhile. But the noise during shutdown couldn't be that. The shroud was a good guess, also maybe the little belt tensioner shock. Also, when the motor mounts sag a bit, the little motor shock absorbers next to the mounts will bottom out and click as the engine rocks. I had this for nearly a year before I found it.

Good luck,
DG