Watch what happens if we simply change reference frames:

time
^
| * * * *
| * * * *
| * * * *
| * * * *
+------------------------ position

time
^
| * * * *
| * * * *
| * * * *
| * * * *
+------------------------ position

time
^
| * * * *
| * * * *
| * * * *
| * * * *
+------------------------ position

time
^
| * * * *
| * * * *
| * * * *
| * * * *
+------------------------ position

Every galaxy sees the other galaxies moving away from it at a rate
proportional to their distances.

True. In fact, Hubble's law is nothing more than Distance = Rate *
Time: You can verify this by checking the units.

Hubble's Constant * Distance = Velocity

50 km/second per MegaParsec is equal to somewhere around 1/13 billion
years.

The diagrams you have drawn show a Galilean Transformation, showing a
fairly small change in speed, less than ten percent of the speed of
light. This would cover the area within a billion light years of
Earth; within 10% of the radius of the universe.

When we get outside that range, if Hubble's Law still holds true, we
need to use a Lorentz Transformation, as the Galilean transformation is
only an approximation.

But the d=r*t law remains true not only in the Galilean transformations
you have shown, but is also true with Lorentz Transformations. Observe
the following animation, showing a Lorentz Transformation, similar to
the Galilean Transformation you have shown.

http://casa.colorado.edu/~ajsh/sr/wh...pacetime_wheel

Notice, when the lines are nearly vertical, they are fairly similar to
those you've drawn with ASCII above. The lines at the edges, on the
other hand, are squeesed in extremely tightly.

Though the lines get squeezed in together, they do not change their
linear quality. They are straight lines, indicating a linear
relationship, preserving Distance=Rate*Time.