My left arm and hand are numb and my fingernails are blue. Could this be something serious?

Depends. This may be raynaud's phenomenon, which is a blood vessel spasm. It is classically triggered by cold. Treatments vary from medication to surgery. It may also be related to autoimmune diseases. Best to get evaluated by a physician for a full history and physical. Make note of what seems to trigger this. If it doesn't resolve quickly, you may need an er visit.
Yes. You should seek medical attention immediately at your closest er.