In which O-class does my Θ-result belong?
You've solved the recurrence more precisely than the test-setter was expecting you to.  You know that $T(n)=\Theta(n\log n)$, which means that $T(n) = O(n\log n)$ and $T(n) = \Omega(n\log n)$, and that gives you the answer.

(By the way, I didn't check that you solved the recurrence correctly, but you can do that yourself. Answer checking is off-topic here, and you didn't ask for it, so I focused on the rest of your question.)