The norm that you consider, as a function of ##a##, has a jump at ##a=0##. Therefore you cannot find the value at ##a=0## by considering the limit ##a\to 0##.
Yes, but in that case you are no longer considering two photons. And by the way, the expression "Feynman rules" means something entirely different, it's related to perturbative Feynman diagrams, not to Feynman path integrals.
Electrodynamics (interaction of EM field with charged fields) is also nonlinear, but quantum electrodynamics is linear and the superposition principle is valid. The point is that electrodynamics, gravity, etc. are non-linear in the sense that Hamiltonian is a non-quadratic function of fields...
I guess we are talking about nonrelativistic QM, not about QFT. If you have two particles, each moving in 3 dimensions, you can think of it as one particle in 6 dimensions. Write the path integral by pretending it's one particle in 6 dimensions, and you will get the right result.