> Do library authors have to test against every type checker to ensure maximum compatibility?

Yes, but in practice, the ecosystem mostly tests against mypy. pyright has been making some inroads, mostly because it backs the diagnostics of the default VS Code Python extension.

> Do application developers need to limit their use of libraries to ones that support their particular choice of type checker?

You can provide your own type stubs instead of using the library's built-in types or existing stubs.