So, since titles like 'sgt' are not the same across the branches, is an e-5 in one branch of equal rank to an e-5 in another (and so forth)?
In battle, do you have to follow orders from a higher grade officer from another branch if you're together and you're not in contact with your own superiors? Can a high-ranking officer override the orders of a lower officer of another branch, or does each branch act entirely independently of the others (save for potus)?
A rather complicated bunch of answers.
1. In the situation where a group of people are cut off from their superiors, the senior person is in charge. It is that person's responsibility to take charge and it is the subordinates' duty to follow their orders, regardless branch of service. In a combat environment, if that officer has any brains, he's going to grab the highest ranking grunts in the group and ask their advice if the officer is not one.
Military personnel are assigned billets -- positions. Those positions make up a pyramid from bottom to top with each service assigned its role -- in the case of joint operations. Only those at the top will be intermingled operationally with the other services. Going down the pyramid, each unit and person answers to their unit.
At the top of the pyramid is one person. Under him he will have a staff made up of all the services that fall under his command. Those people will speak for and direct their respective branches of service to the Commander.
So, for the most part, each service carries out the orders of the commander independently. It's not a gagglefuck. The Army has its job and they are trained a certain way as are the Marines. We probably couldn't understand each other's lingo if we tried.
It's highly unlikely that military personnel will operationally fall outside their own commands though, and the odds of some senior officer from one branch countermanding the orders of a junior officer are slim. The senior officer would first off have to fill a billet with operational control over the junior officer.
Junior officers don't give orders. Their seniors tell them what they are going to do and they in turn tell their subordinates what they are going to do, but it is rare that a junior officer is in operational command of anything but carrying out a senior's order. So if someone comes along and tries to countermand it, he just calls his CO and explains why he isn't carrying out the mission and lets his seniors resolve the problem.