Been noticing when i cold start, sometimes my idle rpm goes up then dips, goes up, dips....like the engine is trying to compensate for a lack of o2 or fuel. After about 10 seconds it levels off and seems to be fine.

Will drive until it gets to temperature and everything seems to be fine - rpm, etc. I get to my destination and shut off the engine.

This is where i have a problem. After several minutes or even a couple of hours, i would then start up ( the still-warm engine ) and it would seem fine until i'm caught at a long idle. If i am at a long traffic light or stuck in a drive through, it would go limp mode and the engine would run rough. Shut off, remove key fob, restart fixes it. Again, on a cold start and once it gets up to temp, it doesnt matter if i'm stuck at a light or not....it doesnr go limp mode. Only after its been shut off for a while and then reatarted that i get that.
Having said all that, do you guys thing its the:

HPFP
Fuel pump
Clogged intake valves
Injectors
Or something else

Would it even run "normal" if any one of the above is failing?failed?

 

Checking your other threads, you have a 2010 Cooper S? My gut says HPFP...