Is it possible to see games or diagrams in the app? It works fine for me in a browser, but in the official Stack Exchange app it only shows me the moves and no board.
No, this is not possible. Of the many site-specific post formatting features, only MathJax is enabled in the Stack Exchange app. The reason for this is probably that it's lots of work, so they did it just for the one feature which is widespread across the Stack Exchange network.
As a workaround, you can click the 'Share' button, open the post in a browser and switch to the full site if necessary: