Multiple random electrical issues is a common sign of a battery that is nearing the end of its useful life. I'd start by getting it tested at one of those free joints like autozone (if you don't have a voltmeter or a good local shop). - Jake . . 2004 325Ci - 5MT - SOLD & MISSED. 2003 3.0 Z4 - 6MT - SOLD TO HAIRDRESSER.
Quick suggestion, if you car is running poorly now that the weather is cold, replace at least the lower intake elbow, the DISA O-ring and the CCV lower oil return hose, clear the DME codes, about $50. Then worry about other things if the car is still running poorly or has additional CEL/MIL codes. The BMW M3 is a true BMW enthusiast's must-have. I bought this car at 90k miles and have driven it 30k more without many major issues - just one dead gas pedal which cost me ($120, 5 mins to m8Oi8.