Is every problem with an output's size that grows polynomialy np?
No. All problems in $\mathsf{NP}$ are decision problems and, as such, the size of their output is trivially polynomially bounded (the output is just one bit).
Moreover, there are decision problems that are not in $\mathsf{NP}$. Indeed, by time hierarchy theorem $\mathsf{NTIME}(n^k) \subsetneq \mathsf{NTIME}(2^n)$ for every constant $k$.