Android - Check whether an app was sideloaded or installed via play store
[If] I originally installed an app from the play store but then side loaded an update would it say installed through the play store or via a third party


It would say sideloaded. You can check the status through the command (requires adb):

adb shell pm list packages -i PKG   # replace PKG with package name of the app


In output: in case of sideloading, the installer would show null value. If the app is updated/installed through Play Store, the value would be com.android.vending.

I tested this on Android 6.0.1 though.


  [Does] google play services usually update in the background due to being prompted to check when you are updating other apps or is it just a coincidence


Yes, it does so on its own. I have setup a toast to inform me which app is updated in the system, and I noticed that the Play Services often updates in the background when Play Store app is  active on the screen. 

No third-party app can automatically force any other app in Play Store to get updated or be sideloaded automatically. It could be just a coincidence  that the both happened at the same time.