How does the length of the output of a problem inform its complexity?
No, it's not an optimization problem.  It takes at least one step of computation per bit or word of output (depending on the model of computation), so since the output can be $\Omega(2^N)$, that means the worst-case running time will be $\Omega(2^N)$, no matter what algorithm you use.  Thus, the complexity is $\Omega(2^N)$.

Yes, the size of the output is a lower bound on the running time to produce that output.