Though I do not find a reference now, I have heard no non-standard model of Peano Arithmetic has a principal ideal domain as its ring of integers. Is that right? Is it trivial? Or is there a good reference? I have looked in Simpson's SOSOA, and in Hajek and Pudlak Metamathematics of First Order Arithmetic.
That is interesting because fairly weak fragments of PA prove the unique prime factorization theorem, so that conservative second order extensions of PA prove every ideal of integers is principal (here I use the obvious interpretation of integer arithmetic in the natural numbers).
I believe the resolution of this tension is just that the definition of an ideal $I$ in a second order extension of PA implies closure under sum of any internally finite list of elements of $I$ (i.e. any list of bounded length), while the claim about non-standard models invokes ideals which are only closed under sum of genuinely finite sets of elements of $I$. Is that right?
If $Z$ is the integers of a nonstandard model of PA, let $n\in Z$ be any nonstandard positive element and consider $2^n$. Then the ideal $(2^n,2^{n-1},2^{n-2},\dots)\subset Z$ generated by $2^{n-m}$ for all standard natural numbers $m$ is not finitely generated. More generally, any element of $Z$ with a nonstandard number of prime factors gives rise to a non-finitely generated ideal in a similar way.
Another perspective on what's going on is that PA can prove that every "finitely" "generated" ideal is generated by a single element, by induction on the number of generators. But this only applies to ideals which can be generated by a "finite sequence" of integers from the perspective of your model. Some sequences which are actually infinite can be encoded this way (if they can be definably indexed by a nonstandard integer), but not all can. Moreover, as you observed, "generated" here means generated under sums indexed by internal natural numbers, not just ordinary finite sums.
Note also that "unique factorization" as encoded in PA does not imply that the integers are a PID, since the factorization is indexed by an element of your model. This means that an element of $Z$ might actually have infinitely many prime factors (as long as those prime factors can be indexed by a nonstandard element of $Z$). This is exactly what happened in the example above.
As for your final question, that's exactly correct. If you look at the second-order proof that every ideal is principal, it first uses prime factorization to get that any ideal is finitely generated, and then it constructs a generator by taking a certain linear combination of the generators, by induction on the number of generators. But since your prime factorizations only have an internally finite number of prime factors, your "finite set of generators" will only be internally finite, and then the single "generator" of the ideal you find will be an internally finite linear combination of all of those generators, which need not be an actual finite sum.