Under what terms will a judge grant emancipation to a 16-year-old?

Very few US states allow the emancipation of a minor. Those that do establish laws regulating the act, that being the case each state differs to a certain degree. In some states the minor must have parental permission to file an emancipation petition. In all states that allow the act parents have the right to contest it. Basically the minor must prove to the court that they are gainfully employed and are able to provide themselves with shelter, food, clothing, medical care, education, etc. This does not mean moving in with a friend, a significant other or a relative or using financial public assistance. The minor must also file the petition themselves and pay all court costs and attributing fees. The minor must prove to the satisfaction of the court that they can handle their personal and financial affairs without the need for adult intervention. Put that together with the fact that minors cannot enter into legal contracts. They cannot drop out of school without parental permission. That being the case they could not rent an apartment, buy a car, have utilities (electricity, gas, phone, etc.) installed and so forth and then you should get an idea of just where the petition will end up.