Xs are sensitive to cold weather. Do a search on "cold" and you will see.

Most of the cold starting problems seem to come in two flavors (1) not much juice in the battery which causes the X to drop systems it deems unecessary (2) freezing of oil in the Oil Breathing components (again a search will show more and better information).

If I were constantly getting messages and SES during the winter, the first thing I would check would be the battery for dead cells.

As for the Key, sometimes it does not respond at great distances. I wonder if maybe you were expecting it to work from too far away. I assume that you kept trying until you were right up one the car? If so, then as you surmise, maybe the key has died.

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2006 X5 Sport 4.4i