My problem was somewhat similar
in my 95 E420 I had that cold start hesitations and rough idling problem whenever i left the car in the garage for 2-3days undriven and the problem went away after 5-10 minutes. The only difference is that my car exhibited this problem as soon as the engine runs and not 15 minutes later like you describe. Since then, I have done few things and I haven't seen this problem come back yet. Here are the things I did:
1) Replaced vacuum hoses. Those are the small L shape that you can buy for $1.30 each from the parts store. I replaced everyone i could get my hands at. a total of about 8 hoses. Including the ones to/from the fuel pressure regulator.
2) Replaced O2 sensor.
3) Replaced spark plugs
4) Tested and filled battery with distilled water.
Doing all the above gave the car nicer feel and better pickup when moving from a stand still. I am an average DIYer and did all the above myself and didn't take but a couple of hours. Some of the suggestions I received when I first posted my cold start problems were to check my fuel pressure regulator, Coolant temp sensor and the Battery as someone indicated that a weak battery may cause this cold start until it gets full charge and then the problem goes away. Good luck finding your source of problem.