Is there a built-in function to see how many of which mobs I've killed? I've seen such lists, but is it built-in or part of some trigger/programming system?

They'll be part of various systems. Treant has one, and there are certainly other systems with them, although I can't think of any separately provided ones.

There's a zMud script over on the Aetolia forums here. It's fairly simple and should be pretty easy to adapt to any client, given that it's really just a collection of triggers more than anything else.