Majority of apps deleted themselves (including Google Play services) on a Samsung device
The same thing has happened to me.

The fact I also have a Note 3 with exactly the same symptoms makes me believe it is a faulty system update or other service installation/upgrade error.

Google Play Services v7.3 (according to https://play.google.com/store/apps/details?id=com.google.android.gms but maybe it's 9.0.82 or 83 or something) rolled out from May 16, I reckon this is the culprit.

I woke up to my phone complaining the same error: "Get Google Play services / Google Play Music won't run without Google Play services, which are missing from your phone".

I managed to install Samsung Internet for Android via Samsung Galaxy Apps.

I then downloaded a Google Play Services apk after searching Google for one and installed it.

That kinda got Google stuff working again, but I had to reinstall a lot of my apps. Authenticator still works thank god.

Boot still takes ages, and every time Google Play Services is gone again.