1. About Signal, WhatsApp, and other instant messengers, you hit the nail on the head.
rb411 when I installed/setup the app I had GPlay so it was relying on that and didn't switch when I deleted GPlay? Basically; would reinstalling the app take care of that?
Apps tend to check only once on first install whether Google Mobile Services (GMS) are installed or not; if you delete GMS, third-party apps that were installed while they were present will still act like they're there. But since they're actually not, you get broken functionality like missing notifications. Updates won't help either; you actually need to first uninstall the apps completely and then re-install them from scratch.
If you install your messengers without GMS present on the phone, they will opt to use their own internal notification services instead that don't rely on GMS and will continue to use those even if you later re-install GMS. So yes, you can get reliable notifications without GMS. Just note that most messengers will need to be given both unrestricted battery usage on the background and unlimited background data usage rights for them to work properly; some apps will notify you for you to give them, others just expect you to know to do so on your own.
rb411 Notifications; specifically from Signal & Whatsapp:
In my experience using WhatsApp without GMS, it works fine and dandy; calls and notifications get through just fine for me. As for Signal, I'd recommend using the Molly client instead of the official one, as its GMS-free notification implementation is much more battery efficient than Signal's own. The official Signal client does still work, but your battery will not thank you. Notifications and calls work fine through Molly as well.
2. Can't comment on this since I don't use WhatsApp backups; never have, probably never will.
3. Don't really know about this since I've never tried to use Android Auto (AA) when denying Google Services Framework's (GSF)Network permission. I have used AA without any internet connectivity in wired mode though, so I don't see a reason why it would not work with a USB cable and a wired connection. Wireless AA probably doesn't work without GMS, but that's just a guess; test it out for yourself.
Installing GMS shouldn't break your messenger notifications provided that you installed them when GMS were not installed. See answer 1.
4. Obtainium has a list of crowd-sourced app configurations that allow downloading certain proprietary apps. You can find the list through Obtainium's "Add app" page; it's located at the bottom of the screen. Maybe some of the proprietary apps you need could be found in there.
If using the Google Play Store is completely out of the question, then Aurora is indeed the "best" secondary option available in getting stuff installed that's only available there; but since there's not really any competition, it's also the "worst." It has it's own issues which you should be aware of if you decide to use it; just search the forums and there are plenty of arguments for and against it.
Good luck.