It sounds a lot like the well known intermediate levers issue. Even though the symptoms, (misfires and rough idling at cold engine start), seems to happen more often at colder temperatures than you are experiencing.

It is very easy to find out if you have the intermediate lever issue. If your 4.4 N62 was 'born' between june 2004 and february 2005, then you have the issue.
All N62 engine produced in that time frame has it. -Sadly mine too.

It is a design/engineering/manufacturing fault/mistake/brain fart, take your pick, from the producent of the intermediate levers, that makes them shrink in cold weather to the point where the computer thinks something is wrong, and cuts the ignition to the affected cylinder(s).

If you want it fixed i recommend that you prepare yourself for a heart attack when you get the price.
The good news is, that if you can live with these less-than-super-smooth cold starts, then you don't have to do anything. The problem wil not get worse.

A tip from a pro intermediate-lever-experienced person:
As soon as the stumbling/rough idling begins, stop the engine and wait 20-30 seconds before starting it again. -Enjoy the now smooth idling engine and don't think about it until tomorrow morning. ;-)

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2007 E70 4.8i Sport
2004 E53 4.4i Sport - R.I.P.