As far as I understood money myself, your intuition is correct. All fiat currency are credit money, so that when you are holding a $, either in cash or bank deposit, you are holding someone else liability. The system is balanced, so that total liabilities are equal to total assets at any time. The net value of the entire monetary system in the economy is zero.
That’s right, but that’s the private sector as a whole. Some part of the private sectors will increase their debt, while others their savings. Clearly that would generate business cycles/boom and bust, and that’s the big discussion of macroeconomics in what is the role of governments in damping/preventing them.
Since the Treasury owns the Fed, the profits made by the Fed are channeled back to the Treasury. The ECB is a bit more complex, but it works in a similar way. When a central bank buys government debt, that debt is the facto neutralized.
A small caveat is perhaps that fiat currency doesn’t have to be debt based, but in practice seems to always be, thus it’s maybe even a bit unfair to call it “fiat” money because it actually does have something backing it indirectly. I think there might be some evolutionary forces at work here: fiat money that isn’t grounded in something tends to suffer hyperinflation because printing money is just too tempting and so we really only have debt-based fiat currency left after the winnowing process.