Which office has limited budget-making powers but presides over the commissioners court and is elected to a four-year term in a countywide election?

The correct answer is B. The county judge

This isn't areal court but rather a legislative body like a government. It is found in states like Arkansas, Missouri, and Texas. They don't really do any judicial business or if they do it is extremely limited since they're primarily a legislative body. The person that presides over these "courts" is the county judge who isn't a real judge such as those in district courts.