Hi! I think you are starting a great adventure with your car!
I am not a mechanic. I write about your issue as I went through it myself. Twice, actually. And I am basing what I write on a thorough study of the related threads on this forum.
There are a lot of threads on this forum about hot start issues. Type "hot start issue" - exactly like I wrote and you will see many results.
As for the starting aid, primarily cold - there are, I think 3 systems, older and newer on our cars. In essence it is about a temperature feeling device sending a signa to activate a Cold Start Valve on the intake manifold to inject additional portion of fuel for certain time and in certain engine temperature conditions. The same set of devicces is responsible for some injection during warm start, but it may not work during the warm start. My impression is Mercedes have never solved this issue really...
The details of these systems are described in the Technical Manual, in the sectin for Full Members. You may consider becoaming a Full Member - it is not a huge investment, it pays off very, very quickly.
Your system seems to be working perfect when cold.You can briefly check the influence of operation of the cold start system on your hot start by manually providing power to Cold Start Valve while cranking hot engine. Just for e second or two.
Reading through the threads on hot start issue, you may conclude that most or a lot of these issues are related to fuel supply. Engine is in the very lean condition at this moment. Fuel supply need to work perfectly (this short injection of fuel through CSD helps there).
If you briefly check electric side by seeing you have a strong sparks on all cylinders, you may look at the fuel side. Probably the first test is to undo your return fuel hose where it gets to fuel tank, put it in the empty fuel canister, switch on ignition (and fuel pump) and see if you are getting 0.8 - 1.0l of gas pouring within 15 seconds. If you get below - you need to check where the fuel system is clogged:
- fuel filter in the tank (if you say you poured gas in the gas station and you could not start, I am excluding the inner gas tank fuel passages)
- additional (not original) fuel filter you may have between tank and fuel pump
- filter in the inlet of the fuel pump
- fuel pump itself
- fuel line
- main fuel filter
- return fuel line
You just check/clean piece by piece.
Observe safety rules when working with fuel.
If you blow air to the return line to hear it bubbling inside the fuel tank and/or to clean it - cover the opening on your side through which you blow air so that it does not flush back at you in cae the line is clogged.
In my case the hot start issue was caused by:
1. Clogged filter between fuel tank and fuel pump (non original filter that, I think, we need)
2. squeezed fuel line (becauise of my stupidity)
I am sure others will chime in with their advise - probably better than mine.
I think, however, the most time and money efficient way is to go systematically through the components rather than shooting from the hip.