Is it hard to start after leaving it overnight? If not, then you may be experiencing a vapor lock problem. Most likely the fuel accumulator is bad. That shouldn't be causing your stalling problem though.
When was the last time you replaced the plugs? What plugs did you use? Have you looked at the distributor cap lately? How about the rotor? Have you ever replaced the plug wires?
How much history of the car do you have?
Personally, for the starting problem, I'd check the ignition components first since they're easy to test and relatively cheap to replace. Then look at the fuel system.
I'd also check the breather hoses and clean out the idle control valve. This could be one of several possibilities for the stalling problem. You seem to have covered several of the more common causes of this. This thread talks about some of the not-so-common problems:
'87 300E Stalling
Do a search and don't ignore the threads on the 300E. There's a lot of good information in the 300E/M103 threads that apply to your engine also.