Sandboxed Google Play doesn't have any special access or capabilities. It doesn't run in a different sandbox from other apps. Each app using sandboxed Google Play contains the Google Play libraries. Only apps containing the Google Play SDK and FCM library are using FCM. It would be entirely possible for that library to function without Google Play services installed if they chose to implement fallback code running a foreground service and asking for a battery optimization exception. It's clear why they don't do that because it'd be inefficient having each app maintain a connection on their own and goes against how they want developers to do things.
I know that push notifications of many app rely on GPS, and that's the only reason I have them installed, but I never found clear information about what are the other implications of giving network access to the app.
Network toggle controls access to the network either via direct connections, or indirect access through OS components or other APIs which provide interfaces for usage by apps which require the permission. DownloadManager is an example within the OS, which performs downloads on behalf of apps. Browsers are an example which do not generally require the internet permission to open links in a new tab, although they could and probably should require it which is something we plan to address.
The Network toggle is not an overall data exfiltration toggle. It doesn't control non-network-related things. It doesn't stop an app putting data/metadata into a file it saves, doesn't stop it abusing permissions you grant it, doesn't stop playing audio, doesn't control other permissions, doesn't control file access (which is done by other permissions and case-by-case grants) and doesn't control app communication without a profile. It's a Network toggle, just like Contacts is a contacts toggle and doesn't stop 1 app given Contacts from sharing it with another app not given Contacts within the same profile. That's not something specific in any way to Network.
If I deny network access to an app, and GPS runs in a sandbox, how can I now if an app is still sending data through it because it's designed to do so?
It runs in the standard app sandbox, not simply a sandbox. This is part of your misunderstanding. You believe it works differently from other apps, but it doesn't. App communication within a profile doesn't depend on the network and that doesn't mean apps aren't sandboxed. Apps can't access each other's data. Apps can mutually consent to communicating with each other within the same profile. These things are stated repeatedly in what you're linking.
What are the privacy risks of running GPS with only network permission allowed? People on here always get very upset whenever someone asks these "dumb" questions, but understanding the EXACT intricancies of app network activities is not trivial, especially for people who don't work in the field.
No one is getting upset about asking "dumb" questions. This has all been asked and repeatedly answered in depth. Making new threads about it only makes it harder for people to find the existing responses and forces us to post the same answers again and again. It doesn't get easier for people to find the info if there are 100 threads with mostly lower quality answers instead of 1 thread with a very high quality, well explained answered that's therefore easier to find.
You're asking questions already answered by what you're linking and quoting.