I think you should back away from the car long enough to read through the archives in this forum. M103 cold start problems are covered thoroughly... too thoroughly. It could be a bad cold start valve, bad fuel accumulator, leaky injector, leaky fuel distributor or a number of other things. I don't recall reading that a bad temp sensor was ever an ultimate cause of hard starting. So that's my advice - start by checking fuel pressure at various points. The electronics don't do much until the engine has started. Note that I'm not an expert, just someone who spends too much time on this forum.

Sixto
95 S420
91 300SE
87 300SDL
83 300SD