Nice post, but I think you got something wrong. Your structure with a single two-sided infinite chain isn't actually a model of first order PA. If x is an element of the two-sided chain, then y=2x=x+x is another non-standard number, and y necessarily lies in a different chain since y-x=x is a non-standard number. Of course, you need to be a little bit careful to be sure that this argument can be expressed in first order language, but I'm pretty sure it can. So, as soon as there is one chain of non-standard numbers, that forces the existence of infinitely many.