As above, don't just start throwing parts at it. It gets expensive quickly, and may still not solve your issue. If the car is running fine when warm, then there is every chance it will be stepper motor related. Just so you understand what it does, basically it is your idle control - or at least part of it. It is a motor which is controlled by the ECU. It isn't a spinning motor, it is a stepper motor, which means that it can move in increments. As the motor and it's gearing are attached to a lever which acts on the throttle, it can effectively 'push' the throttle, in much the same way that you do with your foot on the pedal. The stepper motor is used by the ECU along with ignition timing to alter the idle speed of the engine. At cold start, the stepper is used more than when warm, to raise the idle speed during warm up. Once the car is warm, it is typically used to compensate for a drop in engine revs at idle due to electrical loading, such as headlights, demister etc, which put more load on the alternator resulting in a drag which lowers the idle speed.
Stepper motors on minis are known to give issues, mainly due to failure of solder joints on the circuit board inside, or sometimes failure of one or more of the coils of the motor itself. The first thing would be to establish if the stepper is working at all. One easy way to tell is that when you have the engine running then switch off the engine, the stepper motor will index so that it is in a good position for the next start. You will be able to hear it doing this, it makes a clicking sound for a couple of seconds. Post back once you have checked this and we will be in a better position to advise.
As for the number of cranks before the car starts cold, I would say that is OK. My 92 cooper SPi also takes about 6 compressions (this will be what you mean) before starting. I initially thought that was odd, as I am sure my last car a 95 cooper SPi started quicker, however I discovered through experimenting that the time taken is due to the time it takes fuel to get into the inlet manifold. It's always the same number of compressions, has never changed so I am not worried about it and would suggest that you shouldn't be either. I find that a hot start is quicker - maybe 2 - 3 compressions.