Why is counting the number of hamiltonian subgraphs $\sharp P $ hard?
I think you could be interested by the counting class Span-P (which is the class of counting problems that can be defined as the number of distinct outputs of a nondeterministic Turing machine running in polynomial time). It was introduced in this paper. Both 1) and 2) are in Span-P. For 1): guess a subgraph, then guess a Hamiltonian circuit, then output the graph (of course order the nodes beforehand so that you don't output the same graph twice). For 2) something similar should work.

In this paper they also show that a version of 1) is Span-P complete (page 375). The only difference is that a number $k$ is also part of the input and they want only the Hamiltonian subgraphs of size $k$ (I'm assuming the "positive integer $c$" was a typo). Also at the end of Section 6 ("A variation of the above...") they specifically mention your problem (so without the restriction of size $k$) and they that they do not know if it is Span-P hard.