Why does my car CD player skip when its cold?
Two things: Many CD players have a shock absorption device that is liquid based, it’s possible that the ext ream cold is affecting the viscosity of the liquid and not absorbing the shocks as it would warmed up. Also the lens of the cd player might be fogging up, same as your windshield fogs when you get into the car, start breathing, and turn on the heat. If you have had moisture issues in your car before this would exacerbate the issue.
Don’t know if it helps, but I thought I had a problem exactly like yours until I noticed it was happening when things got warmer too. Turns out I was on the wrong voltage setting on my cigarette lighter adapter and less juice was getting to the CD player. Doesn’t fit all of your symptoms, but just in case…
Perhaps it’s something to do with the lens transport mechanism – if there rubber belts parts there they could become inflexible, any lube associated with the worm drive/motor could become real sticky, etc. If it skips with both cold, left-in-the-car cds, and toasty warm just-out-of-the-house cds, it could well be the player rather than the cds.