My car has this intermediate lever problem, do I need to fix it?

[bulkwarrior]

I have a 2006 4.4i e53. If the temperature drops to 30ish and under, it will have a hard time start in the morning. The SES light flashes and engine hesitates. Turn the car off, and back on, it gets better. When the engine's warm, the car drives like it should(but the SES becomes steady). If the outside temp is higher, then there's no problem with the cold start, and if I keep driving the car after a couple of days the SES goes off by itself.

I've had this problem for 3 years, and since it costs over $4k to fix it, I just kept ignoring it. During winter months I keep the car in garage which helps a bit, but sometimes when it's very cold outside, the garage is cold too. Plus, sometimes I have to travel to other places and park the car in a hotel parking lot.

Here comes the question: If I don't fix these intermediate levers, will the engine eventually get damaged because of this? If this won't damage the engine, I'll just keep ignoring it. This problem first appeared around 80K miles and now I had 115k miles on the car. I'm hoping to drive another 4~5 years and 60k miles before trade it in. If it won't hurt the engine, then I'll just ignore this bug. Like I said, it only happens in the very cold winter months, and I drive more during summer.

this car is out of warranty long long time ago.

Thank you all for your expert opinion and insight!