N62 4.4L rough idle

[burtsplash]

I have a 2004 X5 N62 4.4L with 135K miles on it. It always start nicely. The problem is when outside temperature is below freezing and as soon as I shift the transmission, 7 times out of ten the engine is running rough and if the outside temp is lower, it's getting worse. The idle is so rough, it's hard for me to get out the garage. At minus 30F, the engine went in the Engine fail safe mode when driving. I had very often the P0300 code and a few P0444. I did change my spark plugs and my fuel filter without any improvement.

Here's my observations:
1- the rough running last for about 2 to 3 minutes and disappears by itself. Sometimes (very few), it doesn't disappears, I shut off the engine, wait about 1 minute crank again and the engine runs smoothly like new. I did that also for the fail safe mode and it worked.
2- less problem when the fuel tank is full
3- no problem when the engine is warm

If you have any suggestion, please let me know, Thanks