This is the wrong "DTS" forum. You belong in the real Deville threads. Moved.
The temp gauge indication will have no effect on idle speed - unless the two symptoms are caused by a loose/faulty engine wiring harness. Study and use the sticky post just above this - "How to pull codes" Your car has its own built-in diagnostic code scanner/reader. Study the proper model/year section of the sticky and practice. Write down all codes found and whether each is Current or History; return to the sticky and open the link with "obd2" in the address - this takes you to a Master Index of all P,B,C, and u code definitions. Write it all down and post it in this thread - don't start a new thread for the same sequence of questions/diagnosis.
Cut out four one foot squares of HD cardboard and make two stacks of two squares each. Drive the car (front wheels) onto the two stacks of squares. Now turn the steering wheel lock-to-lock a dozen times, pausing for a couple of seconds at each lock. That should bleed the steering rack - now shut down and carefully check the level of fluid in the reservoir. Any parts store - GM approved P/S fluid. Cheap.