Sounds like exhaust valve rattle. There is a butterfly valve that opens and closes during certain parts of cold start and normal driving to flow more or less exhaust to accommodate noise and efficiency. It is located on the drivers side near the muffler on the M235i. If you search you will find your not the only one with this issue. I would assume BMW would replace it with no issue. As for the shaking mine has givin a sort of vibration/shaking under cold start and goes away after cold start. May be a combination of the two ?
Good luck
