Negating weights converts maximum path into minimum path problems. This is a common trick that lets you reuse shortest path algorithms for longest path problems. Detecting a negative cycle is not enough. You must verify it affects the path from source to destination.
A cycle in a disconnected part of the graph is irrelevant. Two-way reachability checks (forward from source, backward from destination) ensure the cycle matters. Always check both directions when cycle detection affects the answer.