The Lowenheim Skolem theorem only applies to first-order axiom systems that have an infinite model. So it would apply to the axioms for the natural numbers, yes.

The Godel theorems apply to any first-order axiom system, regardless of whether it has an infinite model or not.

I don't understand what you mean by this. Gödels two incompleteness theorems are about theories of natural numbers, so their models are infinite. I don't understand what you could mean by them applying to finite models.

I stand by my claim. The key point of Gödels incompleteness is NOT that no single theory can pin down a single model, that was known before.