This a great idea! I just wonder how users will react.
"Download our mobile app."
"Give us feedback."
"What's new."
No. If I want to know what's knew, I'll click a "What's new?" link. Put it at the top of the page and make it obvious, but not so obvious that it feels like it wants to move in with me.
Also with analytics you can see how your users are reacting. If you guys have any other ideas, we would love to chat more.