Specific EZL Question

All right, I have included a picture to better explain my problem. Just for reference, the yellow arrow points toward the front of the car (1992 300CE). This is the left-hand side.

1) The red arrow points to a vacuum line. What is its purpose? When the car is running and I disconnect it, the engine sound changes. To me it sounds like the engine gets smoother and becomes less "restricted."

2) The brunt of my problem. When the car is cold (may or may not be relevant) the car will often crank but not start. If I have somebody holding the ignition (i.e. cranking the engine) and I apply the slightest bit of pressure right where the green arrow indicates, the car starts up instantly. What the heck is going on? What exactly do I need to replace? BTW, if I disconnect that plug, the car will crank but not start. Interestingly enough, the car runs fine if I unplug it after the car has started.