Does Bellman–Ford algorithm can handle zero-weight cycles?
Yes.  Bellman-Ford can handle graphs with zero-weight cycles; they aren't a problem.

Intuitively, negative-weight cycles are problematic because they can make the notion of "shortest path" ill-defined: there is no shortest path.  For instance, suppose we have a cycle from A to A with weight -5, and an edge from A to B with weight 10.  Then there is no shortest path from A to B.  There's a path with length 10 (go straight there), a path with length 5 (traverse the cycle once, then go there), a path with length 0 (traverse the cycle twice, then go there), a path with length -5, a path with length -10, and so on.

Zero-weight cycles don't cause this problem.