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I am going to have to take this a little at a time, especially since you seem to be adding unnecessary elements. But we'll see..
 Quote by ghwellsjr It is a frame in which the station is at rest. Along the x-axis we have the following items. (Distance units are in μls and time units are in μs.)
So it is the station's frame, okay..
 Quote by ghwellsjr At x=0, we have Einstein at the start of the scenario. At x=7, we have the location of Einstein when the stop-button is pressed, at t=14. Note that he is traveling at 0.5c.
Okay, being the station's frame, we know that Einstein traveled 7 from x0 during his .5c run because we know that the stationmaster saw the clocks reading 10, thus they had to actually be at 14, when he pressed the button. And we know that the train (Einstein) was at 6 from the first clock at 4 from the button when the button was pressed. The train then was at 10 from the button when it was pressed.

So, I have to assume that you have x=0 at t=0 (which wasn't intended, but I think is okay).
At t=0;
xE (x location of Einstein) = 0
At t=14
xE = 7

So at that point, the button gets pressed (t=14), thus
Button Press == (14,7)S (station frame)

 Quote by ghwellsjr At x=13, we have the first stop-clock. Note that it is 6 μls from where Einstein was when the stop-button is pressed.
Adding 6 to the (14,7)s = (14,13)s as the state for the clock when the button is pressed.

 Quote by ghwellsjr At x=17, we have the location of the station-master and the stop-button.
(14,13)s + 4 = (14,17)s = button state.
 Quote by ghwellsjr At x=21, we have the second stop-clock.
(14,13)s + 8 = (14,21) = second clock state.

 Quote by ghwellsjr I believe this adheres to your description of the scenario.
I think it will do.
 Quote by ghwellsjr [0,0] Einstein at the start of the scenario. [0,17] Station-master at the start of the scenario. [14,7] Einstein when the stop-button is pressed. [14,17] Station-master when the stop-button is pressed. [14,13] First stop-clock when the stop-button is pressed. [14,21] Second stop-clock when the stop-button is pressed.
Verified.
 Quote by ghwellsjr [18,9]s Einstein when the photons reach the stop-clocks. [18,13]s First stop-clock when photon hits it. [20.667,10.333]s Einstein first sees image of first stop-clock reading 18. [18,21]s Second stop-clock when photon hits it. [26,13]s Einstein first sees image of second stop-clock reading 18. [26,13]s Einstein colocated with first stop-clock (when it would have read 26). [42,21]s Einstein colocated with second stop-clock (when it would have read 42).
The orange parts, I don't believe are relevant, but otherwise okay.

 Quote by ghwellsjr I think a little explanation is in order for how I arrived at the events for which Einstein sees the two stop-clocks reading 18. It based on the fact that light travels twice as fast as Einstein, therefore Einstein will cover 1/3 of the distance while the image covers 2/3. The distance between Einstein and the first stop-clock is 13-9 = 4 so Einstein will have traveled 4/3 or 1.333 beyond 9 where he was when the photon stopped the first clock which equals 10.333. The distance between Einstein and the second stop-clock is 21-9 = 12 so Einstein will have traveled 12/3 or 4 beyond 9 where he was when the photon stopped the second clock which equals 13.
Where Einstein was when he saw the flashed stop-time is irrelevant. If he was 20 meters back, he would merely see the same image a little later.

I will get to the rest shortly..