Complexity of Hamilton path in directed complete bipartite graphs
These problems can be solved efficiently.

For Problem 2, both [1] and [2] prove that the problem is solvable in $O(n^{2.5} / \log n)$ time. That is, this is the variant of Hamiltonian path with specific start and end vertices.

For Problem 1, i.e., when the input graph is a semicomplete bipartite digraph, it appears (see [3]) that a Hamiltonian path is found in polynomial-time via bipartite matching.

There's a good amount of research on these type of problems on tournaments and other digraphs. You will find more general results by looking at papers that cite the ones I mentioned.



[1]  J. Bang-Jensen, Y. Manoussakis, Weakly hamiltonian-connected vertices in bipartite tournaments, J. Combin. Theory B 63 (1995) 261–280.

[2] J. Bang-Jensen, G. Gutin, J. Huang, Weekly hamiltonian-connected ordinary multipartite tournaments, Discrete Math. 138 (1995) 63–74.

[3] J. Bang-Jensen, M. El Haddad, Y. Manoussakis, T. Przytycka, Parallel algorithms for the Hamiltonian cycle and Hamiltonian path problems in semicomplete bipartite digraphs, Algorithmica 17 (1997) 67–87.