Lattice Gas Automata and Galilean Invariance

I have been studying Lattice Gas Automata methods (also this), and every time I read up on their drawbacks, I see that they are not Galilean invariant and that the simulations have statistical noise. My question is, why is the case?

How do you prove that a lattice gas method is not Galilean invariant? Every proof of Galilean invariance I have seen is for two particles in different inertial frames, and then they prove that the force on both of these particles is the same. Why does this break for a lattice method? Furthermore, what is the consequence of Galilean invariance being broken? How is the physics affected?

And one final question: What is the reason for statistical noise in such simulations, and where does it take place?