Recently, a problem concerning the equivalence of joint measurability and coexistence of quantum observables was solved (Reeb at al. J Phys A Math Theor 46:462002, 2013). In this paper, we generalize two known joint measurability results from sharp observables to the class of extreme observables and study relationships between coexistence, joint measurability, and post-processing of quantum observables when an extreme observable is involved. We also discuss another notion of compatibility and provide a counterexample separating this from the former notions.