Dear Paul,
Yes it seems to be a broken WordPress installation as CSS and JavaScript is not loading properly and behaving correctly in WP admin. You should consider the following thing.
1. Deactivate all the plugins and see if that sort out your issue. If yes then try to activate plugin only by one and see which broke it.
2. If this menu appears to be similar condition after deactivating all the plugins, then I hate to say this but you’ve to prepare your site from fresh installation and you should consider import/export tool WordPress to import content again to new installation.
