Even more specific here. If you're asking for a bus number, you say "Hányas buszt keres?" Hányas is asking for the number on something. Bus or tram lines, the number on the backs of football players, document numbers. The answer is a number of the same format, ending on -as/-es or similar: egyes, kettős, hármas, négyes, ötös, hatos, hetes, nyolcas, kilences, tízes, and so on. Note how you say "bus ten" or "the ten" or similar, and not "the tenth bus".
Hányadik, on the other hand, happens when you sort things, 1st, 2nd, 3rd, and so on. Runners, American streets, or everything that you set up in a certain order. The format of those numbers is (mostly) ending on -edik or -adik: első, második, harmadik, negyedik, ötödik, hatodik, hetedik, nyolcadik, kilencedik, tizedik.
As for the difference between milyen and melyik, if you ask "what kind of" (milyen) thing you're looking for, you're looking for a general quality. Fast, red, big, small, Belgian. You don't really care about the specific thing, is just has to have this quality. On the other hand, if you ask "which" (melyik) thing you're looking for, you want one specific thing (or one specific group). The red one, the fast ones, that one over there.