I work for an airline, so just a few insights here:
Did it actually say "standby?" You can have a confirmed ticket without having a seat assignment. They hold back anywhere from5 to 20 seats on every plane for assignment at the airport.
I don't work for Delta, but most airlines do lock a set number of seats so that only the agent controlling the flight can assign them (reservations, etc cannot assign them). The boarding pass will say "standby" because that's the way the system sees it, but it doesn't mean that the flight is oversold, just that we have to manually assign you a seat. It gives us some wiggle room to deal with any special issues... kids separated from parents, mobility issues, etc. Dunno if this is the same situation, but it sounds like it from what you describe.
If they actually are overbooked, they have to make lots of offers before they can leave without you. If you hold a confirmed ticket and are involuntarily denied boarding, they have to pay you cash money and put you on the next available flight in whatever seat is available, even if it's first class.
Correct, if you have a "confirmed" seat on a flight, the airline has to pay out big bucks if they deny you a seat, so they will typically try to get volunteers to cover the number of oversold seats. I don't have much personal experience with this, because my airline doesn't oversell, so I only deal with this in an equipment change situation, but it's typically much cheaper to offer a big credit to a volunteer than pay out a denied boarding claim.
She said to check back in 20 minutes and she would be able to start the standby seat assignments (for whatever reason).
She was likely waiting for check-in cutoff. Once the window for check-in has passed, anyone who missed their check in is out of luck, and their seats open up. It just gives the opportunity for better seat options.
They moved us around inside of 1st. They found no indication of an equipment change. I think that someone with more status wanted a seat one of us had booked.
I would guess that you were joined by air marshals.