Models of PA inside a computationally weaker theory

Question

I have two questions about the computability power of theories, one about models of ${PA}$ and the other about the model theory itself:

1. Is it possible to create a model of ${PA}$ inside a weaker theory in that all recursive/computable functions are not representable?
2. How much of ${ZFC}$ do we need to develop a semantics theory? I mean a standard theory that says ${X}$ is a model of ${Y}$, in which we interpret other theories. Does this semantics theory necessarily represents all recursive functions?

Edit: I assumed that a theory in 1 is always some kind of set theory in which we can construct an algebraic structure to be counted as a model. I also assumed that a standard semantics theory has enough power to represent all natural numbers (for example, to be able to enumerate variables of language), so we can define representability just like the definition of representability of recursive functions for arithmetic.