We continue the study of the recently-introduced C123-framework, for (simple) graph problems restricted to inputs specified by the forbidding of some finite set of subgraphs, to more general graph problems possibly involving multiedges and self-loops. We study specifically the problems Multigraph Matching Cut, Multigraph d-Cut and Partially Reflexive Stable Cut in this connection. The last may be seen as a Surjective Homomorphism problem to a path P_3 in which both leaves are looped while the interior vertex is loopless. We consider also another family of Surjective Homomorphism problems to a cycle in which only one vertex is loopless. When one forbids a single (simple) subgraph, our first three problems exhibit the same complexity behaviour as C123-problems, but on finite sets of forbidden subgraphs, the classification appears more complex. While Multigraph Matching Cut and Multigraph d-Cut have the same classification as C123-problems, already Partially Reflexive Stable Cut fails to have. This is witnessed by forbidding as subgraphs both C_3 and H_1. Indeed, the difference of behaviour occurs only around pendant subdivisions of nets and pendant subdivisions of H_1. We examine this area in close detail. Our other Surjective Homomorphism problem, ostensibly somewhat similar to Partially Reflexive Stable Cut, behaves very differently when the input is restricted to some class that is H-subgraph-free. For example, it is solvable in polynomial time on any class of bounded degree. Also, its hardness will never be preserved under any form of edge subdivision.