It is disputed. German judges have held that the answer is "no, the EU is not a state, just an transnational organisation" but the EU openly holds that the answer is "yes".
No it doesn't, and that's not what your link says.
The "EU position" (if there even is a single position that can be called that) is that the EU's treaties are binding on the signatory states, and the EU's court is ultimately responsible for deciding the meaning of those treaties. Of course, if a country disagrees with the court, it is free to leave the EU completely.