This is definitely a very powerful feature and will give sleepless nights to most app marketing…
Abhishek Joshi

Not sure if I understood your last requirement, but GA tracks all non-Play store installs/APK installs already. If you go to Acquisition > App Marketplace, you will see something like this image below. Only issue is it can’t read the the App Store names hence it shows (not set) in most cases but gives you an idea on overall APK installs. You can further segment and track all activities coming in from those APKs