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To answer my own question: It turns out that the self-term element can just be skipped when the sum (approximation of the integral) is computed. This is because the $\hat{n} \times$ removes the component anyhow. A practial implementation confirms that the results are useful.
Your question is many questions. Let me see if I can be of some help for the first two: It appears so. These equations should suffice to trace the projection of the orbit of the ray of light in the equatorial plane. I think this problem also appears in the $a=0$ case (Schwarzschild) and it is related to the fact that you cannot observe anything, light ...