Are there any tests that a doctor could carry out to determine if someone is an alcoholic?

Yes. Yes, there is a dsm IV category for alcohol dependence and alcohol abuse. There are also amounts drank per day or week that can indicate a risk for problems related to alcohol. The hallmarks of any addiction are loss of control of the use of the substance and negative consequences from the use ot the substance, trouble with the law, troubled relationships, loss of job, withdrawals, tolerance etc.
Sometimes. There are a number of medical tests which can suggest long-term effects of heavy drinking, but the diagnosis of alcoholism is based on what alcohol is causing that individual to do. Some new tests, both blood and urine, can suggest levels of drinking days before, and a number of older blood indicate alcohol's effects on bone marrow, the liver, and several other things. But history is important.