Hi Rafachus,
Yes, if you purchased the X-Plane 11 digital download on X-Plane.com, your product key will allow you to use both X-Plane 10 and 11 or either one. When you start X-Plane 10 demo, click through the buttons to enter your product key and it should unlock the full simulator.
For the X-Plane 11 errors, you can try running the installer and picking "update" to fix any missing or corrupted default files. Make sure you install to the desktop to avoid any permissions issues.