> Could one derive the Earth's rotation?

Yes, existing gyroscopes are sensitive enough for that. In 1966, one of the first Soviet attempts to launch the Soyuz failed on the pad because the guidance system remained active after a scrubbed launch attempt. After a couple of hours, the Earth had rotated enough such that the guidance system thought it was off-course, triggering the still-armed launch escape system.

> The motion of the solar system?

This system, probably not. This class of system at lab scale, maybe eventually. It seems that the galaxy's gravitational acceleration on the solar system is on the order of 10^-10 m/s (https://www.sciencedaily.com/releases/2020/12/201203133906.h...), so about 10 pico-g. The article says that cold atom interferometers have current sensitivities in the nano-g range but could still potentially improve.

> By watching vibrations could it be a microphone?

With that kind of sensitivity, making it not be a microphone is probably the harder task. However, the sensitivity doesn't necessarily make it a good microphone, since detecting a small signal is useless if the signal-to-noise ratio is too low.

No because velocity only makes sense as relative to something else. There’s no ‘fixed points’ in the universe

This device will track acceleration rather than velocity

> Could one derive the Earth's rotation? The motion of the solar system?

It's not a motion sensor; it's an acceleration sensor. That's an important distinction because the primary problem with IMUs is that they work of changes in velocity, and error in those measurements slowly accumulate - so they almost always need external references to correct for accumulated drift.

You don't need anything nearly so fancy for a laser microphone.

For the first two questions I wouldn't think so because it detects acceleration. But I also only have a high school level of physics knowledge so maybe I'm wrong.