I am the third child in my family. Which question will directly lead to the answer 'third'?

There is no common, easily understood way to ask this question in English.

What is your birth order?

is probably something like what you mean, but only some native speakers will understand what you mean by "birth order."

Although male primogeniture has been a long standing concept in English common law and culture, the concept of birth order, by number, does not go much beyond that. Children can readily identify themselves as eldest (oldest), youngest (baby of the family), or someplace in between (middle child), they might have to think for some time before they could number themselves.

Who's the youngest ?

Will the the youngest child please sit a the card table during Christmas dinner. Twenty two relatives showed up.

I see three children. Who's the third born.

Who is third in line to take over the family business ?

What comes between second and forth?