Will my favorite app work on the phone?

Each operating system has a different level of compatibility.

GrapheneOS (Completely De-Googled): Every app available on F-Droid will work, since these apps are fully open-source. Proprietary applications like those available on Aurora may or may not work depending on their dependence on Google Play Services.
Since 2022 GrapheneOS has implemented an option to of installing unprivileged isolated Google components. Learn more here.
These components can be installed via the system App store, called 'Apps'. With these installed it is actually possible to get functionality of the majority of apps depending on Google Play Services.

CalyxOS (microG): All apps available on F-Droid, and the majority of proprietary applications available on Aurora will be supported with microG’s replacement for Google Play Services.
Lookup compatibility for specific apps: https://plexus.techlore.tech/