An expert on computational number theory will be able to provide much, much more detail than I can, but the short answer is that as far as anyone knows, primality-testing is the rare exception, and almost all of these problems are hard (for a classical computer)! That is, the detailed properties of the prime factorization, like whether there are more than two prime factors, whether there's a repeated factor, etc., are not known to be in P and are generally believed to have the same order of difficulty as factoring itself (even where explicit reductions aren't known, as in most cases they aren't). For more details, you might start with the book Algorithmic Number Theory by Bach and Shallit.
(Note: There are, of course, a few easy ones, like testing whether N is a prime power! But I wonder if there's some general conjecture to the effect that no property of the prime factorization is in P, unless the property is "degenerate" in such-and-such a sense.)
