Google Play Store - com.android.vending
Google Contacts Sync - com.google.android.syncadapters.contacts
Google Play services - com.google.android.gms
Google Services Framework - com.google.android.gsf
Google Partner Setup - com.google.android.partnersetup
Market Feedback Agent - com.google.android.feedback
Google Calendar Sync - com.google.android.syncadapters.calendar
Google Account Manager - com.google.android.gsf.login
Google Backup Transport - com.google.android.backuptransport
You can enable these, so you don't have to use something like GMSInstaller/GMSActivator, which could potentially be unsafe. If you did use one of these but want to use the stock apps instead, you will have to reinstall Remix OS.
This went unnoticed because there is an app called "JideAppPolicy" that disables them at startup. Apparently, this is the only thing it does. Also, apps disabled in a specific way do not appear in Settings.
Enter the root terminal (Alt+F1) and run this:
pm disable com.jide.apppolicy
pm list packages -d
pm enable <package name>
To return to the GUI, use Alt+F7.
If the Play Store doesn't work immediately, check your date/time settings, and force stop + clear data for:
Google Play Store
Google Play services
Google Services Framework
Then restart Remix OS. You should be able to download apps.
To stop automatic updates, turn them off in Settings > Update > Options. Or, if you want to be extra cautious, run this command:
pm disable com.jide.autoupdater