Possible. It can make you feel the cold more than the sciatica in a way tricking your brain into believing the sciatica pain is gone. If that works, wonderful. Likely it will only last until you warm up at which point the pain may return. If you are not getting relief, I would suggest seeing a pain/spine specialist for other options.
Doubtful. Can't hurt, but no medical benefit i would surmise from the treatment. Potentially, if cold enough, could create an anti-inflammatory effect. However, i believe hypothermia would be the more likely outcome if that cold.