A month old thread - are you still interested in inputs? Or have you discovered the problems?
If you have a 2.0L engine, this you must have a Celica All-Trac w/Turbo? That was the ony US Celica that had a 2.0L engine back in 91 (the others were the 1.6L and the 2.2L naturally aspirated engines).
The cold start rattle is, unfortunately, a problem the 2.2L 5S-FE engine developed over time due to the oil filter mounting and today's oil filter seals - oil drained back out of the head and filter relatively quickly on this engine and, over time, block bearings wore enough that cold starts had too much clearance for no oil pressure until the filter filled and the passages filled with oil, so the result is a 1-2 second cold start rattle. You might be able to run a long time like this, but the rattle is an indication of bearing wear in the block.
The cold start rattle has no effect on an engine shutting down after it warms up. To figure that one out, would need a lot more information from you.