For notifications you need to install all three apps: Google Services Framework, Google Play Services, and Google Play. The GrapheneOS Apps app will install all three even if you try to install just Play Services because of dependencies.
Important that you install all of them before other apps so notifications are "registered" correctly.
And, no, you don't have to log in. Actually, I sometimes log out and later log in again and my notifications work regardless so I think they're tied to a device ID, not a login.