I was speaking in general. If this is about Sync specifically I see a "Show a reminder to update when available" options under General in the settings that may be related.
This doesn't fix your problem but what I believe you are seeing is something asking you to update Google Play Services , this isn't related to the Play Store.
GPS contains libraries and other things apps need to function.
An oversimplification would be that it is an analog to Android System Webview which enables web browsing in most apps not based on Firefox.
Maybe. Go into Play Store store settings(android app settings) -> open by default and turn off open supported links. Not sure it'll work for you but give it a try
Strictly speaking, that is a solution. Fair point though, I forgot it had limited support. Still, there are other alternative OS out there you could look for.
edit: I guess fuck me for having a fallible human memory, sorry everyone i learned my lesson i'll never mention alternative os ever again. also how is the "foss" part being ignored. fdroid? alternative stores? degoogle? hello?