does SHA* collide more or less than a random function?
If we could prove anything one way or another about this, it would be a remarkable result worthy of publication in a cryptography journal, since—except for length extension issues in MD hashes—we usually expect these collision-resistant hash functions to behave like good little random oracles.

Of course, there are other families of functions which are not collision-resistant but from which random choices of function have bounded collision probability, namely universal hashes like Poly1305 and GHASH.  We can use these as building blocks for fast authenticators or PRFs.  But I don't think you were asking about those.