Do local governments have powers other than those grants to them by their State?

Answer 1

The local governments only have power over township, municipalities, counties, school districts and special districts. This is called the Dillon's Rule.

Answer 2

This depends heavily on the nature of the state constitution. Some state constitutions are like the US Constitution, in that they set out responsibilities (and powers) to the state government, and assume everything else belongs to the local governments, while other are considerably different, and assume all power rests at the state level.

In terms of systems of government, most states follow the unity model. A few states which could be construed to be more of a federation still lean heavily in state government's direction as far a division of powers.

In fact, the state is where the lowest level of sovereignty rests - local governments have no sovereignty.