No, the Opera app does not run JavaScript itself. The app communicates to Opera's server, and it's the server that does HTTP with the website, runs the JavaScript, and compresses the HTML and sends it down to the app.
The restriction on downloading and executing code has been in the developer agreement since the beginning. It's not a recent thing.
The restriction on downloading and executing code has been in the developer agreement since the beginning. It's not a recent thing.