The name General comes from the fact that these cities are ruled under the general laws of the state, in this case, general Texas law, and not under specific municipal laws. These gives general law cities very limited power. They cannot raise taxes, they cannot add additional charges in utility bills in order to finance public services such as the police or education, and they practically cannot impose bans.
Sharecropping is when African Americans and poor whites were hired to tend to someones crops. The sharecroppers owned some of of the supplies and got a portion of the crop.