Learn Turkish: Ne hâlde olduğunu sordum. - I asked what state he was in.

In Turkish, indirect questions are treated as noun clauses. The verb sormak (to ask) takes the accusative case for the specific thing being asked (the direct object). The entire phrase ne hâlde olduğu (what state he is/was in) is the direct object of sordum, so it takes the accusative suffix -u (with the buffer n), becoming olduğunu.

Could this sentence also mean 'I asked what state YOU were in'?

Yes, it is completely ambiguous without more context! Olduğunu can be broken down in two ways: as olduğu (his/her being) + n (buffer) + u (accusative), OR as olduğun (your being) + u (accusative). You would need a pronoun like onun or senin earlier in the sentence to know for sure.

Why use the -DIK form (olduğu) here instead of a regular past tense verb?

To embed a question like 'what state he was in' into another sentence ('I asked...'), you cannot use a regular conjugated verb. Instead, you use the nominalizer -DIK attached to olmak (to be) to turn the action into a noun phrase. It literally translates to 'I asked his being in what state.'

Why does the word hâlde have a circumflex (^) over the 'a'?