How do they measure (not just calculate, if I understand this correctly) "a distance less than a millionth of the size of a single atom"? That sounds very difficult to do with equipment that is presumably made of atoms.
You can use "low-coherence interferometry" to measure tiny signals that would be undetectable otherwise. Combine a "reference" beam with a "signal" beam and you get a measurable interference pattern, even when the magnitude of the signal beam is miniscule.
This is what a real-life interference pattern looks like (I just acquired this from an actual interferometer illuminating a painted metal surface):
https://dl.dropboxusercontent.com/u/30682604/fringe_signal.p...
This is now an established medical imaging method (Optical Coherence Tomography) to create 3d scans of biological tissue. It can also be used to measure distance or elevation changes on a surface of anything from a micrometer-level scale to a planetary surface. All you need is to use light with the right wavelengths and two measurements "arms" of roughly the same length.
Yes, it will, but that is already represented in the interference pattern.
They're not measuring the absolute distance to the mirror. If they were, you are right about how the precision would be limited.
Instead they're using the interference pattern to measure a change in the distance measurement over time. So even though the distance is somewhat of an average over many atoms, as long as it is the same mirror, it will be the same average at the same distance.
Because the interference pattern represents photons interfering with each other, its precision is limited by the size of photons--which are much smaller than atoms.
Even so, measuring gravity waves requires ridiculous amounts of precision in the construction of the interferometer. I'm working at the 10^-6 scale, where optics can still be adjusted by hand. They are working at the 10^-21 scale - the sheer engineering challenge is awe-inspiring.
Basically fire two lasers at right angles to each other at mirrors, and see the pattern when they bounce back.
For example their electromagnetic fields may be shifting depending on distance to another atom or another field.
PS: I am no physicist.