My '79 280CE (Euro) with M110 engine has developed a severe cold start problem. This came gradually after I had my mechanic adjusted the fuel mixtures and we raised the fuel milage from about 7 miles a gallon to 14 miles a gallon.
Our suspicion is that the cold start injector is not functioning, not sure if it is a connection issue or if this is plugged or broken. Once the engine runs, it runs very smooth and once hot it starts without any issue, but idles rough.
Anyone familiar with such a problem? Any suggestions for my mechanic?

The only experience I have with this is the cold start injector on a fuel injected 1977 VW bug.
Seeing as how these are both German cars from the same time frame they may use a similar system.
My Bug has the Bosch L-jet system. You can find the detailed manual online.
If you can identify which yours has you should be able to dig it up and print it.
My cold start injector is connected to the base of the intake manifold. So look there.
The cold start injector would have a temp sensor somewhere near it. You need to check both for voltage etc..
If it is similar to my VW there will be a multi-relay near the computer.
Forgive me if I am wrong but I think that has something to do with it.
This may also be controlled by a multi relay.

I really don't have much more than very basic knowledge.