If a pitcher makes a fielding error, does the impending run count as earned?

An error by the pitcher is still an error. A run scored by a runner who teaches on an error is not an earned run.

But how an error affects whether other runs are earned or not is more complicated.

No it does not. An error by a pitcher is still an error.

Some will argue that it should, but they would be incredibly misguided. The idea of calling a run earned or unearned is to have an accounting of a pitcher's ability to pitch, not to field.

For determining earned runs a pitcher is considered the same as any other fielder. So his error is treated the same.

No it is still an unearned run. Some would like to change that stat. Once a pitcher throws the ball he where's a different hat and becomes a fielder.

No, an error is an error even if it was his.

No