A. False If the graph contains a negative-weight cycle, then no shortest path exists for any of the vertices on this cycle and for all the vertices which have some path from source going through any vertex in this cycle.
B. TRUE. If the shortest path is well defined, then it cannot include a cycle. Thus, the shortest path contains at most $V − 1$ edges. Running the usual $V − 1$ iterations of Bellman-Form will therefore find that path.
NOTE that After the termination of Bellman Ford Algorithm. even if there is Negative-weight cycle present in the Graph, we will have Correct Shortest-path weights for those vertices for which Shortest path is well-defined in the Graph. For other vertices also, we will get some value but that won't be a correct value.