Complexity of "does a directed graph have <=1 Hamiltonian cycles?"
Your problem clearly belongs to $\mathsf{co\text{-}NP}$. An easy &quot;no&quot; certificate consists of 2 distinct Hamiltonian cycles. The certificate can clearly be checked in polynomial-time.
The same certificate is also a &quot;yes&quot; certificate for problem in 3 (i.e., &quot;Does a directed graph G has at least 2 Hamiltonian cycles?&quot;), showing that it belongs to $\mathsf{NP}$.