In the Matching Cut problem we ask whether a graph $G$ has a matching cut, that is, a matching which is also an edge cut of $G$. We consider the variants Perfect Matching Cut and Disconnected Perfect Matching where we ask whether there exists a matching cut equal to, respectively contained in, a perfect matching. Further, in the problem Maximum Matching Cut we ask for a matching cut with a maximum number of edges. The last problem we consider is $d$-Cut where we ask for an edge cut where each vertex is incident to at most $d$ edges in the cut. We investigate the computational complexity of these problems on bipartite graphs of bounded radius and diameter. Our results extend known results for Matching Cut and Disconnected Perfect Matching. We give complexity dichotomies for $d$-Cut and Maximum Matching Cut and solve one of two open cases for Disconnected Perfect Matching. For Perfect Matching Cut we give the first hardness result for bipartite graphs of bounded radius and diameter and extend the known polynomial cases.