Yes, most olive trees are frost resistant, but do not like it to get too low. Usually the most cold-tolerant varieties can take temperatures of down to 15-20 degrees Fahrenheit. If it gets colder than that in your area, or stays cold for extended periods, try moving it indoors to a sunny window during the worst parts. But once it's used to indoor temperatures, it can be damaged more easily by sudden placement in a cold environment. Try not to water too much during cold periods because the plant slows down and can't handle as much; too much water can cause root rot.
On the flies, I'm not entirely sure what they are, without a picture, but if you put a layer of dry material (such as pencil shavings or shredded paper) on top, it will discourage the laying of eggs.