Hello and welcome to the forum.

Use the Search function on this forum and find what your codes mean. Once you have that information, you can narrow it down quite a bit.

I suggest testing your oxygen sensor voltage using a multimeter. Again, the search function can be used to give you the procedure and the voltage specifications a healthy oxygen sensor will return.

The reason I suspect your oxygen sensor is that you say the car starts and operates well when the engine is cold, but as it warms up and switches over to the closed loop cycle which uses the oxygen sensor (among other things) it begins to have problems.

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2006 BMW M5 "Heidi" @ 109,000 miles
2005 MBZ C55 AMG "Lorelai" @ 165,000 miles
1991 MBZ 300E "Benzachino II" @ 165,000 miles
1990 MBZ 500SL "Shoshanna" @ 118,000 miles

(On the hunt for a good used M103 engine as of 6/10/23, PM me if you have one to sell!)