Everything is right except Coast Guard is most definitely military. Being under DoD isn't a determining factor for being military. CG has been shifted around a lot, but a big reason that CG isn't under DoD is because they have jurisdiction to board foreign vessels without it being seen as an act of war. For this reason Coasties can be sent aboard Navy vessels, and in the event of a boarding situation the CG flag can be raised instead of the Navy flag and the CG takes control of the boarding.