A graph has an Euler circuit if and only if all of its degrees are even.
If both $m$ and $n$ are even, then $K_{m,n}$ has an Euler circuit. When both are odd, there is no Euler path or circuit. If one is $2$ and the other is odd, then there is an Euler path but not an Euler circuit. So, options A and B are TRUE.
Every cycle in a bipartite graph is even and alternates between vertices from $V_1$ and $V_2.$ Since a Hamilton cycle uses all the vertices in $V_1$ and $V_2,$ we must have $m = n.$
As long as $|m-n|\leq 1,$ the graph $K_{m,n}$ will have a Hamilton path. So, options C,D are also TRUE.
So, all the given options are TRUE here.