'03 cooper s backfire idle and cutoff problems

backfiring doesnt always occur when the rpm plays but the missing filler cap light always comes on right after or during backfire

You need to be sure it is backfire, if it is could be a bad crank sensor or ect. Also bad injectors can cause backfire but you need to read the codes. Also make sure if the explosions don't come from the intake because that is a different issue. When you have more info write to be of more help.

perhaps the engine is remaining in "open loop", meaning it is always reading 32 degrees, and the temp sensor isn't telling the computer, or computer isn't communicating with the i.a.c., so sensors to check .ma.p./maf (same sensor come companies refer to it as either, engine temp sensor...either will cause all these problems, it sounds like the engine is operating as it should but again the fuel is not reducing (as it hits closed loop), 900-1100 is started cold, a few moments, and should drop down...i'd start there...let me know if i can help you further

Did they pull any faults out of the vehicle? I see they have replaced you LDP (leak diagnostic pump) I don't belive your check gas cap light is related to the running rough concern. That pump for the evaporative emission runs off of engine vacuum and if the engine is running rough while the vehicle is trying to test the system it could cause a false reading. I would try a fuel pressure test. It kinda sounds like your engine is being starved of fuel. Hope this helps keep us informed.

mini is a bmw company now, and they (BMW) had to recall i forgot how many thousands of cars due to fuel pump issues, that was 2 years a go or more,