The first time I try to turn on my 2008 R56 Cooper S it is quite rough. Sometimes it turns off about twice until it settles off and everything runs normally. Here's a video of this morning. I drove it a bit, got some groceries, and started up with no problems. Where I live, it is warm all year. Oil level is fine. It is catless, but these are the OBDII errors.

 

You say the car is catless but it appears the engine has not received a tune to take the absence of a converter into account. Otherwise I would not expect to see a P0420 error code nor the other codes related to post converter O2 sensor readings.

Think you probably should just put the exhaust system back to factory.

Yeah, the P0030 error signals a bad #1 O2 sensor. So you could replace that sensor clear the codes and road test the car to see what (if any) subsequent codes appear.

FWIW I had a car (not a MINI) manifest a bad sensor, a heater circuit error, with CEL coming on at engine start. But there was no signs of any engine misbehavior. I was some distance from home and my dealer so I drove the car nearly 400 miles with no issues -- other than the CEL coming on after engine start after fueling the car -- and had all 4 sensors replaced. The car had 132K miles and I figured best to replace all sensors. After the new sensors installed the CEL was gone but to my surprise the engine did run better.

Still I think it too much to hope for that a new sensor will address the starting issues but I have my fingers crossed the new sensor is the complete cure for what ails the engine.

 

I installed the downpipe about 2 years ago and the engine has gone a whole overhaul with new CP forged pistons and rods, Supertech Intakes, and engine block machined. It had been running fine for almost 1,000 miles. It just started the rough cold start now. After it reaches about 195F it runs fine. Went to get groceries and ran fine.
I plan on tunning the ECU in the next few months.

 

My gut says fuel system issue, but if you’re net getting any fuel codes, who knows

 

I have seen this with a bad fuel pump on a JCW.

 

So more fuel will be needed ! The JCW I mentioned is a friends which had a new fuel pump installed and failed after about 1000 miles. He found a used stock one and has not had troubles since. A quality, affordable, easy to purchase fuel pump seems to be a bogie we have not solved !