Putting up for discussion: should this one be better translated as "What is the number of that red car"?
Yes, that might be a better match for hányadik. Sadly, there is no word like "which-th" in English to ask about ordinals.
It is asking about ordinal numbers though, so even your proposal might be a tiny bit off the mark. Like... this question with hányadik is most likely asking about which position the car is in, in an ordered sequence like standing in line or crossing the finish line. The answer would be an ordinal number like ötödik ("fifth") or whatever.
For things that are actually numbered (like buses, trams, hotel rooms, or football players in numbered jerseys) there is a different question, hányas?, which matches your phrase even better. And the answer takes a corresponding form of the number like hetes or ötös ("number seven" or "number five" bus or whatever).
"The how-manyeth" (how-manieth?) is one of the least bad English equivalents, I think.
One could also ask "which number is that red car". That might be a better fit still. It's good enough that I suggested it in a report.
It doesn't address your observation of the difference between "6-os busz" and "6-odik busz", however. That one seems to be a distinction that Hungarian makes, but that English doesn't.
Yes hanyadik is definitely asking for the position of the car i.e. in a race is it first, second etc. The best translation i can think of is What position is the red car in? Which-eth or how many-eth is a good way explain to am English speaker what hányadik means.
As is, it can mean anything. it requires a reference to a numbering system, like what is proposed below in this discussion: What is the number of, or which number is... or which ordinal number is. How manieth or any other spelling does not exist according to dictionaries. I checked and found some forums where this item was discussed. http://forum.thefreedictionary.com/postst10020_How-Manyth.aspx
This must be one of those concepts we would not often use in this context in English.
I get the fact that it is referencing a "number" or "position" but that does not mean you need to put the word "number" into the sentence. It is simply assumed. In English one might say, "Which is it?" The answer could be, "The last one." "It came in last." or "The red one came in first, second, or third place."
Why can it not be, "Which is that red car?" or "Which one is that red car?"
Hányadik is specifically asking for a number answer like first, second, or third. Just the word "which" isn't really specific enough to convey that.