Yes, you can setup pages in your app that don't require authentication. It works exactly like you'd do with Forms Auth.
On #1: in a claims based model, your app relies on an external system to authenticate users and receives evidence that the user is valid in the form of a token. You can completely automate this (using WIF and config files), or you can explicitly trigger the authentication process. In any case, your app won't be responsible for validating legitimate users anymore. It is a responsibility that it delegates to the STS (e.g. ADFS). That's why apps are called "relying parties".
I'd suggest you read the first couple chapters of the A Guide to Claims based Identity for a better understanding of the underlying principles.