Will Play Store update a SideLoaded APK
As of 2015 Google Play will detect apps with the same signature, but they will not be updated unless the user un-installs the app, then reinstalls from the Play Store.
So in short, to answer your questions - no Google Play will not update the application if it is installed from somewhere other then the Play Store.

see:
https://stackoverflow.com/questions/14746330/android-sideloading-applications-and-keep-them-up-to-date-via-google-play#answer-28135853