Is every problem with an output's size that grows polynomialy np? I am wondering if every problem with an output's size that grows polynomialy is $\textsf{NP}$?
My thinking is every $\textsf{NP}$ problems can be solved in polynomial time by a non-deterministic Turing machine. (Wikipedia)
Moreover, consider a non-deterministic algorithm, that takes in input an integer $n$ and an integer $k$. This algorithm randomly outputs  an $n^k$  bits combination. For example $(2,2)$ could output $0101$
Intuitively, this algorithm as a polynomial complexity.
A problem with an output's size that grows polynomialy could be solved by this algorithm.
Therefore, a problem with an output's size that grows polynomialy could be solved by a non-deterministic algorithm in polynomial time.
Therefore, every problem with an output's size that grows polynomialy is $\textsf{NP}$.
I do not know, if my thinking is correct.
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 $k$.