Showing that the language of graphs and nodes on an odd cycle is in NL
The question is ambiguous. Specifically, it is not clear what "different vertices that appear in that cycle" means. It can mean two things (1) the size of the set of nodes that appear in the path, or (2) the total number of nodes in that path, i.e., the length of that cycle (minus 1).

If the interpretation is (2), then the language is clearly NL:
you perform a random walk (of length $\le n$), counting the number of nodes you see. If you get back to $v$ and the number of nodes is odd, you accept, otherwise you reject.

If the interpretation is (1), I don't see an immediate way to show it is in NL.