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2

As you say, non-degenerate codes have a lot of well-understood machinery that's brought in from the classical side. That helps us from a conceptual stance, and a mathematical one (making rigorous results easier to prove), although, in terms of practical implementation, doesn't necessarily mean that one is easier to implement than the other. Errors in a ...

5

The terms $II$ and $ZZ$ do not uniquely specify the state $|11\rangle$ because you could equally have the state $|00\rangle$. Indeed, you should not include the identity term in your stabilizer. Thus, you need to add a second term, which could be either $-ZI$ or $-IZ$. Either way, you can easily see how to make a product $-ZI$ out of your stabilizers.

1

Assuming the encoder itself doesn't have to be fault tolerant, you can do the proof constructively. Prepare the code space by projecting each of the stabilizers into the +1 eigenstate. For each stabilizer $S$: Measure $S$. If you're in the -1 eigenspace of $S$, apply Pauli gates to flip the stabilizer (but not other stabilizers). E.g. stim can compute ...

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