My impression is that you're raising topics far beyond just support for Webauthn and FIDO, most of which I'm unable to answer.
quasarsunerisvenus937 Working: Profile with Google Play Services with minimal throwaway Google Account)
Not working: Profile without Google Play Services
Google Play services is required for practically most FIDO functionality. This is because there is currently no native support for FIDO in the Android Open Source Project (AOSP), which GrapheneOS is based upon. Apps can provide FIDO functionality without Play services if they include a non-Play FIDO library for that, but very few apps do this. Most apps choose to use the Google Play FIDO library, which depends on Play services. Chromium also has no native FIDO support.
quasarsunerisvenus937 I was wondering if there are any thoughts, plans or already existing integration regarding WebAuthn support without Google Play Services in GrapheneOS?
The GrapheneOS project has expressed that there are plans to develop support for FIDO2 in GrapheneOS without a reliance on Google Play services. There is an open GitHub issue on this in the issue tracker. A GrapheneOS developer also recently expressed some potential aspects of the planned feature:
it's very unfortunate that there never ended up simply being FIDO2 via the built-in secure element, that's something we can implement ourselves though
we're going to make our own but it will probably start out just using the secure element for it and not bothering with external keys until later if at all
Note that the GrapheneOS project rarely announces ETAs for the launch of specific features, and there is no ETA for this.
Please note that I do not speak for the GrapheneOS project.
quasarsunerisvenus937 The university explicitly requires a Google Account on Android devices for their WebAuthn implementation, likely due to Platform Authenticator API of FIDO2/WebAuthn requirements (but I'm not sure since I'm not from IT).
Using passkeys with Proton Pass doesn't actually require you to sign in to a Google account in the profile where it's being used. Only the presence of Play services is required. Naturally, if you are to use Google's cloud-synced passkey service you will have to sign in with a Google account.