I get hand and wrist pain when i wash my hands with cold water. is this normal?

No, may be Raynaud's: Raynaud's phenomenon is a symptom caused by constriction of small blood vessels, usually in the fingers and toes, leading to a lack of nutrients and oxygen going to the fingers and toes. Raynaud's phenomenon can be due to auto-immune disorders, hormonal imbalances, medication effects, etc... A primary care, endocrinology, or rheumatology doctor can look for raynaud's and other possible causes.

Not unless it's : Really cold. You may have raynaud's or beurger's. Arthritis more likely relieved by cold. If you smoke, quitting may resolve the issue for a time at least. If you have one of those diseases above and don't quit, you'll start to lose fingers.