The easiest way for you to quickly and reliably access your paid apps would be to use Sandboxed Google Play. If you worry about privacy towards Google, you can set up a user profile specifically for your paid apps and only use Play Services there.
I really recommend to go that route, but if you demand an alternative way without Play Services, you can try installing the app in Aurora Store (logged in to your Google account). For me this used to work for apps I bought directly in Play Store, but not for in app purchases. You might need Play Services to use the app anyway though...
Regarding the APK file: That sounds to me like the file is corrupted, but I can't really judge that with the given information. The recommendation applies: Save yourself the trouble of manually installing (and updating) APK files and use Sandboxed Google Play instead. It's a marvelous feature!