Is it possible to get a false positive alcohol reading if you did not have any alcohol?

Yes. If you are refering to urine tests that look for metabolites, it is possible but not likely. Could be excessive use of alcohol based hand sanitizers. One case i'm aware of is using vanilla extract in alcohol base put in coffee, though using it in cooking should get rid of alcohol. Possible using alcohol based mouthwash, should be avoided any way in someone with history of alcoholism. Nyquil.
Yes. There is a chance but is extremely limited. Eating lots of fruits everyday can lead due to fermentation to alcohol formation. But the concentration in your blood will be extremely small.