We investigate the complexity of finding prime implicants and minimal equivalent DNFs for Boolean formulas, and of testing equivalence and isomorphism of monotone formulas. For DNF related problems, the complexity of the monotone case strongly differs from the arbitrary case. We show that it is DP--complete to check whether a monomial is a prime implicant for an arbitrary formula, but checking prime implicants for monotone formulas is in Logspace. We show PP-completeness of checking whether the minimum size of a DNF for a monotone formula is at most k. For k in unary, we show the complexity of the problem to drop to coNP. In [Umans 2001] a similar problem for arbitrary formulas was shown to be complete for the 2nd level of the Polynomial Time Hierarchy. We show that calculating the minimal DNF for a monotone formula is possible in output-polynomial time if and only if P=NP. Finally, we disprove a conjecture from [Reith 2003] by showing that checking whether two formulas are isomorphic has the same complexity for arbitrary formulas as for monotone formulas.