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To answer your questions at the start, everyone will see those clocks reading 18 μs. Since there is nothing you need to do relativity wise, as Einstein's frame of reference never comes into play. All you need to know is that you press the button when it's reading 10 μs. And it takes 4 μs for the pulse to get to the clocks so they stop plus it takes the light 4 μs to get from the clock to the station manager so when the manager actually hits the stop button he is 4 μs behind the clocks.

 Quote by James S Saint I understand that the other lengths are all CONTRACTED, not lengthened. But do we agree that at the button press moment, the train is at 6 μls for both frames?
That is not possible. I assumed that the information given at the beginning was from the stations frame of reference. In the stations frame of reference you have a point on the track that is 6 μls away from the first stop-clock. This is when the button is pushed. In Einstein's frame of reference that distance is length contracted by .866 so in Einstein's frame of reference that distance is .866*6 = 5.196 μls away. Also in Einstein's frame of reference the distance between the stop button and the clocks is 4*.866 = 3.464 μls. Edit: but since in that frame those points are moving it takes differnet times.