Gödel's first incompleteness theorem states roughly that "for any axiomatization of arithmetic, there are statements that can neither be proven to be false nor true."
Does this still hold when it comes to quantifier-free statements?
I.e. if we have the structure of arithmetic: $(\mathbb N, +,\cdot, 0,1)$, and we restrict to sentences $\Phi^{QF}$ about this structure that don't contain $\exists, \forall$, can we then for all $\phi\in \Phi^{QF}$ either prove or disprove $\phi$?
What you are referring to is called Baby Arithmetic ($\mathsf{BA}$) in Peter Smith's book An introduction to Gödel's Theorems.
$\mathsf{BA}$'s language [has] one single individual constant $0$, the one-place function symbol $S$, and the two-place function symbols $+$ and $\times$. [...] it lacks quantifiers and variables.
$\mathsf{BA}$ is complete, negation complete, and decidable.
I would say that any quantifier-free sentence $\varphi$ is true in $\mathbb{N}$ must be provable from, say, $PA^-$, since it can only be a boolean combination of things like $t_1(n_1,\dots,n_a)\leq t_2(m_1,\dots,m_b)$, where the $n$'s and $m$'s are just expressions of the form $1+\dots+1$, and it can be seen quite easily that weak theories (like $PA^-$) decide all of them.
i think this would also be true for formulas where only bounded quantifiers are used (this follows for example from absoluteness of $\Delta_0$ formulas by end-extensions of $\mathbb{N}$).
