ENGINE CUTS OUT WHEN COLD

Under the airfilter assembly and part of the Fuel distributator.
rarley the problem usually a vacuum related issue in mercedes 124s.
see my other post on 300E rough Idle. THis care uses a hybrid bosch mechanical electronic fuel delivery system. easy to diagnose and assumptions are usually wrong. try w124zone for basic information.

Engine is difficult to start when cold

1. Check the battery condition, battery state of charge and starter motor current draw. Replace the battery if marginal and the starter motor if the current draw is beyond specification.
2. Check the battery cable condition. Clean the battery terminals and replace corroded or damaged cables.
3. Check the fuel system for proper operation. A fuel pump with insufficient fuel pressure or clogged injectors should be replaced.
4. Check the engine's tune-up status. Note the tune-up specifications and check for items such as severely worn spark plugs; adjust or replace as needed. On vehicles with manually adjusted valve clearances, check for tight valves and adjust to specification.
5. Check for a failed coolant temperature sensor, and replace if out of specification.
6. heck the operation of the engine management systems for fuel and ignition; repair or replace failed components as necessary.

Engine is difficult to start when hot

1. Check the air filter and air intake system. Replace the air filter if it is dirty or contaminated. Check the fresh air intake system for restrictions or blockage.
2. Check for loose or deteriorated engine grounds and clean, tighten or replace as needed.
3. Check for needed maintenance. Inspect tune-up and service related items such as spark plugs and engine oil condition, and check the operation of the engine fuel and ignition management system.

you may want to try haveing the fuel injectors cleaned they can also cause miss fires also try flushing the fuel rail to flush the rail just hook up fuel gauge and turn the key on and remove the gauge from the hose and let it run into a container also look at fuel and see if the quality is good should be strong smelling one cup should smell up a small garage if it doesnt try a different brand of gas

Check cold start valve, and thermal timer switch

change plugs and fuel filter

sounds like the fuel pump! have to test it and the relay! could be hunderd or a few bucks.

Aee thw fuel system.

I had this once, get the car running and turn the ignition key very slightly and very lightly from side to side while the engine is at idle, this may show up a faulty ignition switch if the engine cuts when you're turning it.
Also the ecu on the nearside inner wing may be at fault, see if you can find a friend with the same car to try swapping over to his first to test as they are expensive is this isn't the problem.
Finally, a common fault with these cars is the fuel pump relay, its the biggest black plastic covered relay in the fuse/relay box, at the rear of the front offside inner wing area.

I would start by replacing your automatic air intake and throtle body sensor. Check these 2 things out and let me know.

tepfy

I had a similar problem with my 1986 300e , which got progressively worse until the car was no longer driveable, I changed out the fuel pump relay and the problem went away.