When using adaptive optics, what is the shortest timescale of atmospheric changes in refractive index that astronomers have to deal with? I am interested in the astronomer's need for speed in wavefront correction, especially when measuring in NIR and visible domain.

What is the fastest abberation coming from atmospheric turbulence in any measurement setting (sun observation during day, night-time observations)? Does any astronomer need correction in the visible spectrum beyond e.g. 100 Hz?
Are fast systems practically limited by spatial resolution of the sensor?
Is there any way to get a quantitative overview how widespread adaptive optics is in astronomy today and which setups are currently in use?

Thank you for your thoughts!
I apologise, I'm posting an answer because I can't comment.

What is the fastest abberation coming from atmospheric turbulence in any measurement setting (sun observation during day, night-time observations)?

It depends on the astronomical seeing conditions at the location of the telescope, and also varies quite a lot. It could be anywhere from $1-100~ms$.

Does any astronomer need correction in the visible spectrum beyond e.g. 100 Hz?

It depends on the what you're after. In solar physics, for example, observations are generally interested in resolving features on the Sun. You can also find polarimeters that operate beyond $100~Hz$, so an Adaptive Optics system that operates in the $kHz$ is a fairly standard requirement for most modern Solar telescopes. However, if you're only interested in collecting unresolved spectra (say, a star), you may not really benefit from having a really fast Adaptive Optics system.

Are fast systems practically limited by spatial resolution of the sensor?

I'm not sure what you mean by limited and sensor. Are you talking about the sampling of the wavefront by the wavefront sensor?

Is there any way to get a quantitative overview how widespread adaptive optics is in astronomy today and which setups are currently in use?

I think you'd find this reference helpful: https://link.springer.com/article/10.12942/lrsp-2011-2