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All tensor products of $n$ Pauli operators $\{I,X,Y,Z\}$ (that is $4^n$ combinations) form an orthogonal basis for the vector space of $2^n \times 2^n$ complex matrices. Hence, for every matrix there is a unique decomposition as a linear combination of tensor products of Pauli unitaries. Same is true if we fix some other unitary basis. If we not fix the ...
First of all, are the ancilla qubits entangled with $|\psi\rangle$? Yes, depending on what the state is. Let's say you started with $$\alpha|000\rangle+\beta|111\rangle,$$ but it has experienced an error of $(\cos\theta I+i\sin\theta X)$ on the first qubit. So, your state becomes  \cos\theta(\alpha|000\rangle+\beta|111\rangle)+i\sin\theta(\alpha|100\...