Your question very closely mirrors this question, and I highly recommend you look into it.
The simple answer to your question is that saying ～だか is somewhat unnatural, as you suspected. Like @snailplane mentioned in the comment below, ～だか does happen occasionally if something is being left unsaid. i.e. ～なんだか（わからない）
~のか is the question form of ～のだ、and here we see that the だ is omitted when answering a question. The same holds true for pretty much every other case, except subordinate clauses as discussed in the answers (second answer) to the linked question.
However, in informal speech, you'll find that often Japanese will omit saying か、with a rising intonation at the end of the sentence that says, "this is a question." This is something that you'll have to listen to a bit to get the hang of. So, if you're fairly new to the language, I would recommend using か until you get the hang of things.
Read both of the answers to the linked question, because they do a very good job of explaining this concept.