1223 is definitely the coolant temperature sensor. And yes, a faulty one can definitely cause driveability issues. I'd replace it and have a shop that has a BMW scanner (or you can even borrow one from someone) reset the code. It's a 20 dollar part, and I'd bet money on you being able to find someone near you that has a correct tool to reset the light (if not, any shop that does have it will do it in 30 seconds ... and I really doubt they'd even charge you anything for it).
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