Start by shutting off the breaker in the main panel that feeds the subpanel.
Using a multimeter set to check continuity, test the continuity through the main breaker in the subpanel. It's possible that the main breaker in the panel is bad, and needs to be replaced. The problem is, it's usually cheaper to buy a whole new panel then to get a replacement main breaker. If this is a new panel, and the breaker tests bad. I'd contact the place you purchased it from, and ask them what your options are.
While the power is still off to the subpanel.
Check all the connections to make sure they are tight.
Check the continuity from the bus bar through each branch circuit breaker.
Make sure the main breaker in the subpanel is fully closed. If you turn it to the ON position and it has a little wiggle to the handle, this might mean the breaker is not actually setting when you turn it on. In the ON position, the handle should held tightly in place. If the breaker will not set, it means the breaker is bad and needs to be replaced.
I'm fine with the idea of a main panel with only 2 things in it feeding a sub-panel with everything in it.
Question #1 - I'd rethink putting a service panel outdoors at all. Weather is rough on panels, even if they claim to be outdoor rated. I'm a little nervous about a 100A breaker supplied from the normal bus bars, but if the manufacturer stands behind it, okay. The 100A wires are going to be a mother to wrestle onto that 100A breaker. Are you quite sure the power company has provisioned you 125A service? 100A is more common.
Question #1 (the second): You're gonna want more slots than 24, since this box powers pretty much your whole house. Nobody ever installed an addition and went "Gosh, that job was sure made harder and more expensive by having too many slots in the panel". It's a false economy, especially since bigger boxes are often bundled with more breakers. Your house may be ok now, but do a kitchen remodel and lookout!
Question #2 (the second): Don't bond your grounds to random plumbing that happens to be going by. It's not code, and someday you might have a plumbing problem and the plumber replaces a downstream chunk of it with PVC. Whoops. Also, they've been upgrading customers to PLASTIC water meters. Double whoops. Bond properly and to code.
Question #3 (the third): Bond ground and neutral only in the (singular) main panel. As such, you need 4 wires between main and subpanel.
Just for your edification, it's only a sub-panel if it's fed from a main panel. If it's fed directly from a transformer, it's a main panel.
Best Answer
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.