Java is no longer supported so the 2 errors when starting the trainer cannot be helped anymore. It's just a unsupported browser script that sends you to the site when you click the window on the trainer itself. As for the main issue, you may want to start a support ticket, as I don't own this title and am limited to just basic support. Not sure if Steam will auto open and run through Ubisoft Connect now, being one of the earlier games to use Uplay and Steam. Steam might just need it's own trainer now.
www.cheathappens.com/support.asp (scroll to bottom to open a ticket)
Cool. The Java thing makes sense. Just for giggles I uninstalled the Steam version and installed the UPLAY version. It was only $7.50 so I figured I'd give it a try. But it actually works even less... After you activate the trainer at the main menu then start or load any game, the whole thing just exits out or crashes with no error message. Oh well, guess I'll go back to playing something else. Anno 2070 is really old anyway (but fun).