Can the Play Store (in /system/priv-app/) be reinstalled with a newer version?
Yes and No, it depends on the version of thd play store and the Google play services, both usually work together and can not work with the wrong version, so try to replace both apps... 

You should use OGapps ( Open Google Apps ) , it's designed for custom ROMs.

You might have problems with the version designed only for DATA installations, and may need an alternative app from another custom ROM that is the same version.