leaving the nice, quiet purr of the classic gasoline V-6 engine that 300E's are famous for.
That sir, is an inline 6 cylinder engine, not a V6.
Problem 1 MAY be the air pump operating. It's belt driven, mounted over near the alternator. I can't tell you if there is a problem with your s or not (if the noise is normal or not) but the air pump runs on cold starts.
Problem 2, could be lots of things: worn plugs, incorrect plugs, carbon on the valves, a leaky fuel injector or two, bad gas or low octane gas, coolant temp sensor.......
Could also be valve stem seals, although the "false start" with valve stem seals are usually upon cold start.