"What did you buy her for her birthday?"
Translation:Doğum günü için ona ne aldınız?
Doğ- is the root for "being born". It's also used for the sunrise. In Turkish, the Sun doesn't rise; it is born every day. (Güneş doğar).
The place where the Sun "is born" every day is called "Doğu" (East).
Similarly, at dusk, the Sun "sinks", from the verb root bat- (to sink). The place where the Sun "sinks" every day is called "Batı" (West).
No, compound nouns that are in the format of "X - Y[+ 3rd person possessive]" (like, doğum gün-ü) need to be relieved of this possessive ending before "being owned" by someone else.
So you revert it back to "Doğum gün" and then add further possessives:
Benim doğum günüm, senin doğum günün, onun doğum günü.
İ think it is because 'ona' is remaining and it would be 'onun doğum günü'. Your sentence says ' What did you buy for her/your birthday'. I am not sure if Duolingo would accept the sentence as 'onun doğum günü ne aldın' But I think that without 'ona' this could mean for example, that you wıll buy new clothes for yourself in order to look good at the birthday party or you will buy some flowers for the mother in order to thank her for the invitation to the party. Your sentence does not mentıon the person to whom you are goşng to buy something.
Yes, I think so. You can say "Ona doğum günü için ne aldin." The question word has to stand where the given answer is standing. Let me give you an example: For her birthday I bought her a book. "A book" is what you are asking for. It is replaced now by "ne", but as the verb nearly always is standing in the end of the sentence it would be "ne aldin". My English is not the best, but I hope I could make it clear.
Your assumption that both mean to "for him/her" is not quite correct. They signify different kinds of objects, similar, but not identical to the difference between direct and indirect object in English. Ona is "to him/her" or "for him/her" - dative case. Usually, but not always corresponding to an indirect object using a preposition like "for/to". In this sentence "ona" corresponds to the first "her". "What did you buy her for her birthday", so that is an example where you do not have a preposition in the English form. "Onu" is "him/her" - accusative case. This - again usually - corresponds to a direct object without a prepositon in English. Not so in your example, however - "I waited for her after school" is more like "I awaited her after school" in Turkish.