Just got my first Benz. I've been a long time admirer. I grew up with a 72 280SE 4.5 in the family that mom still owns that was purchased new by her dad from Sutherland motors in Spokane. Always thought it was an impressive car considering when it was made.
Anyhow, enough shouting the praises.
I've got an 89 300TE and I've purchased a repair manual that is less than astounding, so it's time to ask the knowledgeable.

I've noticed in this hot weather that parking the car and shutting it off after it is fully warmed up makes the temp gauge read hotter when I start it. I noticed there is a small electric coolant pump in the engine compartment and wondered if it is supposed to run after the engine is shut off. Turbo Audi's had a similar setup that is supposed to run after shut down and the pumps fail frequently. I just wonder if this is a similar scenario.

On a recent road trip I noticed a small amount of play in the steering while driving straight down the road. In other cars that I've experienced that have a steering box, as opposed to a rack, there was a stud with a locknut on the top of the steering box that would allow a person to adjust out that slop. Will I find that on my W124 steering box? If not, what should I be looking for as the culprit? BTW, this car has @ 172K miles.

I'm not new to wrenching, but I am new to Benz's. You may have gathered by my username that I'm also a fan of another german make.
Many thanks for your help.

__________________
Fourings

84 Audi Coupe GT with Turbo engine, EFI (buh-bye CIS), Jamex seats, Koni's/H&R's, e-code lights. 210k and counting.
1992 Anthracite 400E, 161k, "Grey Ghost".
1986 VW Quantum Syncro, 192k, gone to the crusher in the sky.
1989 300TE 172k. (Was mine, now also Mom's car)
72 280SE 4.5 (Mom's car)