If the gauge simply stays past cold, I think its the gauge of the sender unit or a loose connection in the back of the speedometer.
My temperature gauge does this as well, however, when it is at "cold" when it should be at hot, I use two hands and equal pressure and slightly push the whole cluster up and back (toward the engine) and immediately the temp gauge works perfectly. I had read somewhere (Sorry I don't have the time to find the link) that the cluster on the e36 'sags' and in some instances this makes the gauge faulty. There are some DIY fixes which use common garage items to make sure it doesn't happen, but that requires removing the cluster, which requires removing the steering wheel (there are some that say you can fit the cluster throug the big holw of the steeringwheel, but that is VERY difficult and it really is less work to take off the steering wheel and doing it the correct way), and I have not had time to go through all that effort to fix the minor issue. If your cluster is not sagging and this does not do the trick, it is most likely the temp gauge itself or the sending unit.
I am not sure as the idle pulley. In regard to the 1000, 700 step down, mine does that only after I've been driving it hard. I would have someone check out your idle pulley, and if it is messed up do it yourself. It's really not difficult because there's so much space to work because it's not the L6 motor.
May be post the idle pulley question in the e36 forum; you'll probably get more responses on it there.
Good luck & welcome to the board!