Is it possible to use query strings in the game page? Something like user.itch.io/gamename?foo=bar and get the embedded html5 game to see it? Or just being able to set them in the options of the file to embed would be nice.

You should be able to access the URL of the parent itch.io page via your javascript  and extract the URL parameter from there. We don't have an official API to access that data yet.