Can someone provide me a numerical simulation of Feynman path integral? Where the contribution of each path is added individually so that I can understand how much do the paths outside the light cone actually contribute and how they actually interfere destructively with each other?
There are an uncountably infinite number of paths, because spacetime is continuous, so direct computation of the path integral is impossible without some discretization, outside of some very special cases. You might want to look into lattice field theory, which is precisely the restriction of quantum field theory to a discrete spacetime lattice.
Instead of numerical simulations, consider to explore Feynman's checkerboard model for (massive) fermions in a discretized 1+1D spacetime. Much of the "sum over histories" of the FPI and the causal structure of the lightcone can be recovered from purely analytical considerations.