Context-sensitive fusion grammars and fusion grammars with forbidden context are special cases of context-dependent fusion grammars where instead of finite sets of positive and negative context conditions in the first case a rule has only a single positive context condition and in the second case a rule has only a finite set of negative context conditions. In this paper, we show that several decision problems can be formulated quite intuitively by these grammars. In particular, we show that the Boolean satisfiability problem as well as the Post correspondence problem can be transformed into a context-sensitive fusion grammar and that the graph coloring problem can be transformed into a fusion grammar with forbidden context such that the generated languages give solutions for the problems. Furthermore, we prove that both grammar types can generate all recursively enumerable string languages (up to representation of strings as graphs) and are universal in this respect.