What phrase do you use to ask for the n-th position of an object? Is there a straightforward and general way to do so?
e.g. order of childbirth, place did the runner finish
Since you can use 3人目 for third, can you use 何人目 or 第何子 to ask, perhaps?
(This especially hard to ask since English has problems conveying this.)
Yes, you can say 何人目【なんにんめ】. You can add 目 to a counter regardless of whether it's attached to a specific number like 3 or a question word like 何, so you can say things like 何人目 or 何代目. Here's an example of the latter from ALC:
「クリントンはアメリカの何代目の大統領ですか？」「第42代大統領です」
"Where was Clinton in the chronological order of Presidents?" "He was the 42nd president."
More generally, you can say 何番目, which means "what number [in a sequence]". Amusingly, ALC includes almost the exact same example for 何番目:
「クリントンはアメリカの何番目の大統領ですか？」「第42代大統領です」
"Where was Clinton in the chronological order of Presidents?" "He was the 42nd president."
As you note, this is difficult to express in English, so you might use this sort of expression in Japanese when we'd normally talk about cardinal numbers in English. Take a look at the following example from Kenkyūsha's New Japanese-English Dictionary:
あなたの本当の病名を言い当てたのは何人目の医者でしたか。
How many doctors did you see before one guessed the name of your illness?
Note that the Japanese asks about ordinality (what position in the list of doctors), while the corresponding English translation asks about cardinality (the total number of doctors).
