I agree that the accumulator is a likely place to begin, particularly b/c you say that it's only the first start that has hard cranking. A properly-functioning accumulator keeps the fuel system pressurized enough to open the injectors without a lot of cranking. If pressure bleeds off due to a poor accumulator, the fuel pumps have to run a while in order to re-pressurize the system, hence, long cranking times (altho 10 minutes I've never heard of). The quick test is to cycle the ignition key 3-4 times before you attempt to start the cold motor so that the pump runs each of those times and is allowed to build up some fuel pressure before the engine is started. If that test helps, you've identified a fuel pressure problem, almost certainly the accumulator.

To answer your original question, yes cold start valves can go bad like anything else mechanical (I've replaced one on my M103), but I don't think it's a very common failure.

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1988 California version 260E (W124)
Anthracite Grey/Palomino
Owned since new and still going strong and smooth
MBCA member

Past Mercedes-Benz:
1986 190E Baby Benz
1967 230 Inherited from mom when she downsized
1959 220S Introduced me to the joys of keepin' 'em goin'

There are only 10 kinds of people in the world--those who understand binary and those who don't