1) What is it, for a belief to be 'rationally justified'?

Sorry to disapoint you, but this is a very broad topic. Thus, my answer won't be able to do justice to this question. There have been Dissertations, books, and anthologies devoted to just this topic.

This question is analogous to asking: "what is truth?" While the question seems simple enough, when one gets into different theories of truth, one soon finds the answer is far more complicated then one would expect.

In my answer, I will be basing it on several assumptions:

1) A belief is considered justified, if and only if, all/most of the evidence points to it OR it is what we in philosophy call, foundational. The first assumption is obvious enough, however, the second, begs for an explanation.

A foundational belief is that which is justified in and of itself. In other words, it is not justified by any other beliefs. An example of this, would be a belief in a logical axiom. For instance, nothing justifies the law of excluded middle for two valued logic, which states: Either P is true or P is false.

Some have also argued (like myself) that sensory beliefs are justified in and of themselves. Nothing needs to justify sensory beliefs.

Now, everything I am saying, is based on the theory of justification known as: Evidentialism and Modest Foundationalism. Like most things in Philosophy, this is controversal.

If you would like a good book, that adressess issues of what "Justification" is, pick up 'Epistemology" by Richard Feldman.

2) What do you understand by "evidence" when using language like "there is no evidence for God"? Put more abstractly, when does x count as evidence for y?

If X supports the claim Y, then it is considered evidence. In order for something to be considered evidence for a given claim, it must support it. Or, to put it another way, X is evidence for Y, in the case that one can infer Y from X.

Of course, the more evidence for a given claim, the stronger the claim is.

I hope this answers your questions. You seem to like Epistemology and Philosophy of Science. Philosophy Of Science deals with exactly what evidence is, and Epistemology deals with what knowledge, belief, truth and justification are.

If your interested in these topics, you should search for books on these topics.

- Email addressed by Chaoslord, the logician of the squad.