Noticing the 'n' in hatvanharmadik isn't audible. Is there a pronunciaton rule regarding when that happens?
No, not really. You can say it accurately if you want. It just naturally becomes less audible if you say it quickly. In this case, it sounds like the "n" in the word "hang".
I have noticed the inaudible "n" in other Hungarian words, like "minden" -- usually in an internal (not initial or final) position.
Really? I have never thought "minden" contained an inaudible "n"... interesting.