Both are correct. The first one depends entirely on context, because 're is a contraction for both are and were.
In the second set, "I don't believe it" is an expression of general incredulousness. "I don't believe that" is a bit more argumentative, suggesting that you have a different thought or interpretation. Example: "He died" - "I don't believe it!" (you really aren't questioning the fact of his death, just that it's hard to comprehend) whereas "I don't believe that" suggests you think perhaps he is in hiding somewhere, kidnapped, or the like.
Student or Learner