Dev here. First, let me apologize about the mess!
To the best of my knowledge, both the live tile and the ads issue are related to WP8.1 bugs. They affect a handful of users, and trust me when I say I'm as frustrated about them as you are! I believe that the ad issue is caused (for the users that are being affected) whenever a new update is released. Upon first launching after the update, the OS doesn't properly "recognize" the app, and wrongly reports its payment status. It should be the case that if you terminate by repeatedly pressing the back button and re-launch, that the problem goes away.
The notificatoin issue is much more annoying. Send me an email and I'll give you a list of steps you can try for resolving. However I am willing to bet money that a hard reset will resolve it regardless (which by itself as a solution, points to this being an OS, rather than an app bug).
I'm sorry again for this.