We really need to know if the connection is made via metal conduit, particularly Rigid (pipe threaded) metal conduit. The metal conduit is the ground wire, meaning the white neutral wire is not the ground wire.
Any wire which ordinarily handles return current from a hot, is a neutral. Neutral is not ground. Wires which only handle current during electrical failures are safety grounds.
The idea of hooking up an outbuilding without a ground was outlawed some time ago, and for good reason. For awhile, Code allowed a toxic hybrid: "just combine neutral and ground". "After all, they go to the same place in the main panel, right?" But it's a terrible idea that actually makes things worse. If you had no grounds at all, a broken neutral just broke your power. With neutral bootlegged to ground, a broken neutral floats all your grounds to 120V, making things which are supposed to be safe, lethal!
Applying a ground rod is no cure either. Dirt is a poor conductor (which is why we bother digging up copper) and the dirt between shed and house doesn't have the capacity to return large amounts of current like you have in an electrical short. You end up just electrifying the soil as well!
Reminds me of a story about a woman whose phone ringer didn't work. "But I know when someone calls, because the dog yelps". Hooboy.
If you have separate neutral and ground
-- then as Someone says above, you need to have separate neutral and ground buses. Since outbuildings need a main shutoff, people usually use "main panel" type panels and use the "main breaker" as the shutoff. These are intended to be main panels, which don't need separate neutral and ground buses, so this is often omitted. Either they sell separate accessory ground buses, or they let you split a dual netural bus into a neutral bus and a ground bus.
Since many "electricians" mostly fling Romex into new houses or do repairs, they often don't understand the finer points of subpanels. So it's common to see electricians wiring subpanels like main panels, glomming neutrals and grounds onto the same bus. Either grounding wasn't taught when they went to trade school, or they got so habituated to working in 1-panel homes that they forgot. Either way, it's the kind of error that doesn't fail right away.
If you have neutral only
Then you are bootlegging ground, with the risks therein.
You can fix that by retrofitting a ground wire and separate your neutrals and grounds. The ground wire does not need to follow the same route as the conductors. If the conductors are in conduit, you can add the wire. Otherwise you can direct bury a bare copper wire of appropriate size.
Alternately, you can render that safe by putting the entire shed on a GFCI breaker (this being a 2-pole breaker if the shed has 240V). In this case you must separate neutrals from grounds or it'll defeat the purpose! Now, if something bites you, it is returning not through the neutral wire, and the GFCI will detect this and trip.
Another option is convert the circuit to 120V-only. The wire colors may (or may not) be the right colors for this to be legal. In this case, one wire becomes ground, one becomes neutral and the other becomes 120V hot. This is often good enough for people who aren't doing heavy work.
Neutral and ground being separated means you must remove the neutral-ground bonding device (commonly, a screw) from the neutral bar. You must also have a separate ground bar, which the cheapie big-box panels don't always come with.
You absolutely need both local grounding rods and a ground wire from the main panel. They do different jobs. Remember, electricity travels in loops and wants to return to source. The local grounding rod returns ESD and lightning, which is sourced from earth and should be returned right at the shed, not carried 150' on #6 wire to your house. Human electricity is sourced from the neutral bus on your transformer; the ground wire returns fault current to the neutral in your supply at the main panel. Dirt doesn't conduct well enough to do this job, all you will do is electrify all the grounds in your building.
Since you are doing a pole line, you really ought to use a "messenger wire" optimized for carrying the physical weight of the wire. This messenger can double as the ground wire, in fact you will want to earth it!
You are correct that a subpanel main breaker is simply a shutoff switch, which is mandatory in outbuildings. Its ampacity does not matter at all. Playing games with breaker sizes to try to get the nearer breaker to trip first, doesn't really work.
If you use aluminum wire use #4Al instead of #6Cu. They have the exact same current capacities and you will not need to change your calculations at all.
Also stop buying at big-box stores when you get as sophisticated as a pole line. Go to a real electrical supply, they have what you need in stock and they know what they're talking about.
Best Answer
A GFCI breaker does not know or care what happens upstream (elsewhere in the subpanel or back at the main panel. All that matters is that neutral and hot are connected to it properly so that it can detect the difference between them. If there is a ground wire going to the protected device then that ground must be separate from neutral until sometime past the GFCI - i.e., whether the ground is connected to the neutral bar in this subpanel is a code & safety issue but won't affect the GFCI.
So the GFCI should not be a problem.
On the other hand, some things look a little strange about the panel. I'm no pro, but:
As far as a proper ground path, if your conduit is metal end-to-end then that should give you an easy way to take care of grounding. And if it isn't metal end-to-end, you should be able to fish a ground wire (green or bare copper, size depends on the size of the other sizes) through and then separate neutral and ground in the subpanel.