Value of S and r in Fiat-Shamir Protocol
It's the same situation as for RSA encryption. If you accidentally pick a bad $s$, i.e. one that's not coprime to $N$ then the protocol fails. How likely is this to happen? If you hit a bad $s$, then you can find the factors of $N$, hence you can break the protocol - put another way, finding a bad $s$, whether deliberately or by accident is no easier than factoring $N$ and breaking the whole thing anyway. And the whole reason that you're doing a protocol in an RSA group in the first place is because you believe that your $N$ is big enough that it's infeasible to factor.

As for whether the boundaries $1$ and $N-1$ are included, this really doesn't matter either - if your $N$ is large enough then the chance of accidentally hitting the boundary when you sample uniformly from $[1,N-1]$ is $2/N$ which is tiny.