Well, it depends what you are searching for. /e/ is also based on LineageOS but with its own oecosystem like a cloud-service, app-store and so on. Although - AFAIK - their app store and different other apps like the e-mail-client are just forked from different (open-source-) apps…
I have my own NAS and thus I prefer LOS4microG.
I don’t know those other ROMs, though.
microG is somewhat like a replacement for google play services (GPS). Some GPS are replaced by microG-services. Like Maps API and Exposure Notifications - without communication to google servers 
For other thinks like Push Notifications you have to at least activate Google Device Registration. But this is optional (And with a bare LOS you wouldn’t even have the possibility to do so). Here you can see the implementation status of microG services.
If you want to use Aurora Store and want to install some purchased apps, you still have the possibility to (isolated) link your Google account with one specific app. I for example did this with Aurora Store and Google My Business.
No, LOS is shipped with its own recovery. If you update LOS, your recovery will be overwritten too. But the LOS-recovery is perfectly ok for making updates
Correct, that’s one of the things that should be changed in this guide. I’ve recently installed Magisk on a device so I already had to find out it, too: You just have to rename the “.apk” to “.zip” its written somewhere in their FAQs.
Normally, you just have to go to Settings, there you should be able to search for updates. But you can also install it manually.
But you may already have the newest update installed. As of now, the newest version is FP3 3.A.0101.20210420
Sorry, I can’t tell you much about /e/ since I don’t use it.
TBH, I never heard of that. Or I forgot it 
But I can’t recall any recent issues with the FP3+ camera.