Has any astronomical measurement ever been accurate to within less than one standard deviation and can you give examples?

Answer 1

Yes. We know the distance from the Earth to the Moon very precisely, on the order of millimeters or less. The Apollo astronauts left "corner reflectors" on the Moon's surface, and when we fire a laser pulse at it, we can detect the precise moment when the reflection gets back to our sensor. Because we know the speed of light VERY precisely, we can calculate the distance based on the time from when the pulse is fired to when the reflection is returned.

We can calculate the distances to other space probes the same way - we send a radio signal, wait for the reply, and calculate the distance.