My tooth pain happens when eating or drinking cold things?

Yes, that happens. The nerve pain can cause sensitivity to hot or cold liquids or foods such as ice cream. Warm is better, you should see a dentist.
See your dentist. This means the nerve endings of the tooth are exposed and there is probably a cavity on the tooth. See your dentist asap.
See your dentist. Cold sensitivity can indicate a number of things: too much pressure on the tooth (grinding etc), gum recession, infection/cavity, possibly a cracked tooth, or even sinus problems. A dentist can help you figure out the correct remedy.