1983 533i

The following problem is when my car is cold (left overnight). The car is hard to start, usually takes two or three times of cranking for it to catch. When it does catch, it will stumble for approximately 2 minutes. During this initial warmup, if the accelerator is pushed quickly the car will die. It will not rev very high, about 1750 -2000 rpm, and you can only rev it if you are very gentle, pushing the accelerator slowly. The car is driveable after about 30 seconds of idling, at this point the car must be revved to about 2000, and then the clutch can be engaged and I am off. If during the initial two minutes, the car is allowed to slow down i.e. the revs fall below 2000, it will buck and try to die. As long as the accelerator is gently applied, and engine speed is kept up, everything will be alright. Reminder this only happens when the engine is cold.

Problem two happens at all times, when the engine is quickly revved, it will stumble, and then rev up. Also, a puff of air seems to come out of the plastic intake piece next to the headlights. Also, when the engine is at high rpms, and then allowed to fall to idle, the car will almost die and then catch, and idle at its usual idle rpm.

Both of these problems started after a headgasket replacement and duck gasket replacement.

Things I have done to try and remedy the problems.
AFM was switched for a known good one.
The four sensors on the car that measure water temp were all tested, and replaced if needed. These are the sensors that are right next to the thermostat housing.
A new fuel pressure regulator was installed.
A new air boot from AFM to intake was installed.
The throttle plate was properly adjusted.
New vacuum lines were installed.
New cold start valve.

note: I ordered a new main relay but the one I was sent did not allow the car to start at all, a new one has been ordered.

I was assuming that an air leak is causing the hesitation/stumble upon acceleration, and the stumble when the car falls to idle. If it is an air leak, I have checked and rechecked all possible points, I think. It is possible I am missing something somewhere. Any ideas as to where to check would be appreciated.

I am not sure about the cold start issue, I am at a loss here.

I appreciate any and all ideas.

Thank you