Is Gödel's incompleteness theorem provable without any model-theoretic notion?
"A formal sentence being true iff every interpretation of it in every model is true". Not so. A sentence of formal arithmetic may be true on the intended, standard, interpretation while not a logical truth, not true on every interpretation.
"Gödel's theorem is completely formal." Not so. The theorem is a bit of ordinary informal mathematics. Its subject-matter is formal proofs in certain kinds of formal theories. But Gödel's theorem is a meta-result about those formal theories, and the usual proofs involve ordinary mathematical reasoning. [I say "ordinary" proofs because there have been projects to produce formal computer verifications of Gödel's theorem: but the theorem pre-dated the formal verifications by decades. And it is a nice question what we learn from having a formal proof in Coq, for example, all 37906 lines of it.]
"Gödel's theorem does not rely on any model-theoretic notion." Actually as perhaps Mostowski was the first to make really clear in his exposition, sixty years ago, there are two different versions of the first theorem, one that presupposes that we are dealing with a sound theory, another that assumes only that we are dealing with an [omega]-consistent theory. These are often referred to as the semantic vs the syntactic versions of the theorem. The semantic version, unsurprisingly, does rely on semantic notions.
"Are there proofs that use the apparatus of logic but do not not use any model theory at all?" Yes, some results, including the syntactic version of Gödel's first theorem, are purely proof-theoretic, i.e. about syntax. But again note that a theorem's being about syntax doesn't itself make it "completely formal". (For another example, proofs about the normalizability of formal natural deduction proofs in first-order logic are themselves bits of informal mathematics.)
Although it is not usually presented this way, the usual proof of the incompleteness theorem shows the following:
For any sufficiently strong effective theory $T$ of arithmetic, with Gödel-Rosser sentence $G$, there is an algorithm that will take any proof of $G$ within $T$ and turn it into a proof of $0=1$ within $T$, and there is an algorithm that will take a proof of the negation of $G$ within $T$ and turn it into a proof of $0=1$ within $T$.
This has the immediate corollary:
For any sufficiently strong effective theory $T$ of arithmetic, with Gödel-Rosser sentence $G$, if $T$ does not prove $0=1$ then $T$ does not prove $G$ and $T$ does not prove the negation of $G$.
Neither of these statements, nor their proofs, requires any model theory.
However, these syntactic statements leave open the question of whether $G$ is actually true. It's one thing to know that there is some sentence that is neither provable nor disprovable, but either $G$ or its negation has to be true as a statement about natural numbers, and it would be nice to know which one it is. The answer is that $G$ is true, rather than the negation of $G$.