Shor's quantum error correction code with unknown basis
Q1 : Fate of the 9 qubits
There is no (or 2) answer to your question, since it is implementation dependent :

If your implementation of the code is ideal, and you have a way to directly measure the syndromes, the 9 qubits are projected on the global 9 entangled qubits state. It is the case, for example, if the code is used in a fault-tolerant implementation of quantum computing.
If the implementation is more realistic, the measurement of the 8 syndromes is performed in 2 steps : 1st you apply a global unitary (through a quantum circuit) on the 9 qubits, then you measure 8 qubits, destroying them in the problem. You have then a single qubit, to which you can apply a unitary(depending on the measurement result) to get the single original qubit.

Q2: Effect of misalignment
The misalignment behaves like an error, whether $x$ is small or $π/4$ (as in your $\lvert±\rangle$ example). What you describe correspond to applying the operator $\cos x I +\sin x XZ$ to each qubit. Measuring the syndrome projects the state to one with a given number of errors, but as long as there is less than 1 X error and 1 Z error, the code is enough to correct it.
This condition happens with probability
$$(\cos^2 x)^9 + 9 (\cos^2 x)^8\sin^2 x=\cos^{16}x(1+8\sin^2x)\simeq 1-64x^4$$
the last approximation being valid for small $x$.
If the rotation is by $\pi/4$, the error rate is 50% and your code will not be able to correct it.