Is finding the longest path of a graph NP-complete?

First, it is easy to see that the problem is in $text{NP}$. The longest path is a Hamiltonian one since it visits all vertices. Indeed, there is a straightforward reduction from $text{HAM-PATH}$ to it. For details and some special cases, see for example here. Likewise, it is $text{NP}$-complete to decide whether there is a path of length $k$ with the a similar argument: just set $k$ such that the problem is equivalent of solving $text{HAM-PATH}$. Finally, if I understood your third point correctly, it doesn’t make the problem any easier if you in addition require the path to visit a specific vertex $v$. Similar reasoning holds for this case.