The length of the path that light travels from the Sun to any observer on Earth is approximately the same. Otherwise the brightness of celestial objects would change dramatically as they approach the horizon.
Not necessarily. Only the divergence and convergence of rays determines apparent size and brightness.
This point requires some additional explanation:
The same amount of energy (for example, a one degree arc of the Sun in each of two dimensions) covers an area of the flat Earth that is several times larger at mid-afternoon than it is at noon.
Because of this, something has to give: either the brightness of the sun is much lower at mid-afternoon (and I do not mean 20% lower) or the apparent size of the Sun is dramatically reduced, or the speed of light is less than half on the vertical direction compared with the horizontal direction, or all of the above.
Now, if you think for a second that the Sun might produce more light through the borders than through the center, you can explain how the exact same sunspots can be seen from every place on Earth at the same time.