I have recently acquired a 2009 cooper S r56. The price was low for the car with 125k but knowing there were a few gremlins to sort out.
Right out the bat had a few codes I attributed right away to cooling, the CTMP-MOM was reading ***C
P112B ECT below regulating Temp
P0128 ECT out of range
P0117 ECT circuit Low

First things i did was to swap in a genuine Temp Sensor. I was able to clear the codes and started getting readings however it would not get above 50-72*C driving, when parked it will reach 103*C. however as soon as I start driving it will begin to drop back down to 50-60*C and then the engine would not be running as great (assuming fuel map is adjusting) sometimes it would read ***C then all of a sudden a reading would reappear, assuming this was out range?
My question is does this require a special adaptation of the sensor in addition to clearing the code? I tried clearing the code on a stone cold engine, same results.
My other thought is the thermostat is stuck open, and perhaps should have bought a new oem one from the start.

P0303 Cyl 3 Misfire
P3091 fuel low pressure
I swapped coil packs, still am not getting a misfire yet on another cylinder, but assuming this is likely the HPFP starting to fail, or the engine was running rough due to the cooling issue
Thanks for any wisdom to chase and clear these issues