Complexity class (P/NP) variants of Hamiltonian paths problems
Problem A is not a decision problem. This means that by definition it is not NP-complete. 

Problem B is NP-complete. But every NP-complete problem is NP-hard.

Let me remind you the definitions.

NP-complete problem. Problem P is said to be NP-complete if it is in NP and every problem in NP can be reduced in polynomial time to P.

NP-hard problem. Problem P is said to be NP-hard if every problem in NP can be reduced in polynomial time into P.