Welcome to the best Benz site on the Net.
As far as maintaining power to the radio when you work on the car, try doing a search. Donald Swinford, I believe, posted a how-to where you attach a 9v battery at the cigarette lighter. This will keep the radio memory powered while you work.
The A/C could be the A/C compressor cycling, could be a vacuum leak or could the ACC control. (again try doing a search for A/C puts out warm air??).
The rough idle, always. Hmmm...seems that was recently discussed at length. As I recall the poster was given a bunch of different reasons/diagnosis and probable costs. I think he found out something that wasn't terribly expensive and it fixed his rough idle.
Can you guess what's next
Yup...try doing a search for 420SEL rough idle. I bet you find lots of good info. Unfortunately a near new model like a 1986 is not in my repertoire...yet.
Lots of luck, again welcome to the best Benz site on the Net. Someone more knowledgeable in the newer models may be able to give you more info than my simply pointing where it is
'73 280SEL 4.5 (9/72)- RIP
Only 8,173 units built from 5/71 thru 11/72
'02 CLK320 Cabriolet - wifey's mid-life crisis
2012 VW Jetta Sportwagon TDI...at least its a diesel
Non illegitemae carborundum.