The carpal tunnel is a narrow passageway in the wrist that is made of bones on three sides and a ligament (transverse carpal ligament) across the wrist where the palm and forearm meet.
Inside the tunnel are:
* Tendons, which are strong and flexible (but not stretchable) bands of tissue that connect muscle to bone. The tendons are surrounded by membranes (synovial membranes) called tendon sheaths.
* The median nerve, which controls some movement in the thumb and gives sensation to the thumb, index finger, middle finger, and half of the ring finger.
WebMD
and there is an illustration
here
I'm not entirely sure why the cold made the difference. Are you applying the icepacks directly or wrapping them with a light towel? That might prevent some damage to the skin. Otherwise you could alternate with the ice on and off more frequently instead of leaving it for more than 10 min. I'm not sure what the recommended time is. I don't believe icing it would cause a problem. If you are resting your wrist on it make sure that it's not drooping over the icepack - keep it positioned so your wrist is level. You may have only irritated the nerve , even not considering the swelling. Giving your wrist a break, continuing to ice it if that helps, and positioning it so that the bones are as open as possible will probably give you the best relief.
Glad the redness is getting better