If an astronaut murdered their fellow astronauts on the ISS, what exactly would happen next? And under which state’s law would their crime be prosecuted?

In terms of crimes on international waters (close enough to space), the vessel’s original country would be prosecuting them. Either that or the country of origin of the criminal.
I’d also assume that the UN would get involved, maybe the ICJ?

They would be prosecuted according to Space Law

They throw him out and if he shall be tried in the country in which he lands

You’re forgetting about ” Space Force “

I would say most likely the state the rocket launched from.