Could a human beat light in a footrace?

No physical laws are being broken in this thought experiment. If you are concerned with the relativistic requirement "nothing can go faster than the speed of light", that only applies to the speed light goes in a vacuum: $c = 3 \times 10^8$ m/s. The reference to light in that relativity postulate makes it sound like if you could only find a situation where you slowed light down, you could break the laws of physics; not so. A better statement of the postulate would be "nothing can go faster than $3 \times 10^8$ m/s, which happens to also be the speed light travels at in a vacuum." I don't see anyone going faster than $3 \times 10^8$ m/s in this thought experiment, so no physics violations.

As for what the human at the end of the race sees:

He sees a blinding blue light from the all the Cherenkov Radiation from even the slightest charged particle passing through the medium. And perhaps the time at the start of the race. It's exactly what you would imagine since we are talking non-relativistic speeds. What an anti-climactic answer, eh?

There is a concept of "slow light" which is looking at light pulses whose group velocity propagates very slowly. This is slightly different than your clock example, but close enough that you might be interested in it.

In 1998, Danish physicist Lene Vestergaard Hau led a combined team from Harvard University and the Rowland Institute for Science which succeeded in slowing a beam of light to about 17 meters per second...

Usain Bolt can run roughly 12m/s, so we are not all that far off.

(Of course, we are playing some tremendous games with the light beams while doing these sorts of slow light games. Whether this is actually applicable to your specific thought experiment involving a clock depends on what aspects of the experiment you felt were important)

Yes, it is theoretically possible.

For example, you could use two parallel, perfectly reflecting mirrors of length $L$, where $L$ is the distance between point A and point B. Let the distance between the two mirrors be $d$.

Assuming that the ray of light enters the two mirror by hitting one of them close to point A at an angle $\theta$, it will be reflected

$$N \approx \left ( \frac{L}{d \tan \theta} \right)$$

times before arriving at point B, covering a distance

$$l = N \frac d {\cos \theta} \approx \left ( \frac{L}{d \tan \theta} \right) \frac d {\cos \theta} = \frac L {\sin \theta}$$

in the process. Notice that the result does not (somewhat surprisingly) depend on $d$.

Therefore, the time needed to go from point A to point B for the ray of light is

$$t=\frac l c \approx \frac L {c \sin \theta}$$

You can therefore define an "effective speed" $v_e$ for the ray,

$$v_e \equiv c \sin \theta$$

If the speed of a human is $v_h$, the human will be faster than the light ray if

$$v_h > v_e \ \Rightarrow \ \sin \theta < \frac{v_h} c$$

The record speed for a running human (*) is $44.72$ km/h (Usain Bolt, 2009). The speed of light in a vacuum is $1.08 \cdot 10^9$ km/h. You get therefore the condition

$$\sin \theta < 4.14 \cdot 10^{-8}$$

You can see therefore that this is not very easy to realize in practice (and we are neglecting refraction, absorption, scattering, surface roughness etc.).

You also can repeat the calculation assuming that there is a material with refractive index $n$ between the two mirrors.

(*) I want to be generous.