I see in the documentation for component resources that it's recommended to register outputs. What can I do with them once they are registered? Is there a way to access them for other resources that depend on them?

Yes. Registering isn't technically necessary, it happens anyway. It just lets Pulumi know that no other changes are expected. And you can use nested resources any way you like; they're just objects.

You can't depend on the registered outputs as of right now. Instead, use public class members or get-accessors to let dependent resources use any outputs you want to share outside the component resource.