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I've heard 'ですか' all too often in textbook Japanese.
I've personally never heard だ and か used together - only 'の' or a rising intonation.
Eg:
なまえはなんですか？ vs なまえはなんだか？
Is it normal/natural in the second case? How do you properly convey a question in this case if not with か?
なまえはなんですか？ vs なまえはなんだか？
Is it normal/natural in the second case? How do you properly convey a question in this case if not with か?
As a simple interrogative sentence "What is [your name]?", 「[なまえ]はなんだか？」 doesn't sound natural/correct. You can instead say:
[なまえ]はなに？ -- friendly, gender-neutral
[なまえ]はなんだ？ -- relatively masculine, can be blunt
[なまえ]はなにか？ -- not common in daily conversation, might sound strong/harsh/rigid
Similarly, to say for example 「これは[誰]{だれ}の[傘]{かさ}ですか？」 "Whose umbrella is this?" in the casual form:
これは誰の傘？
これは誰の傘だ？
これは誰の傘か？
(×これは誰の傘だか？)
As to yes-no questions, to say for example 「これは[山田]{やまだ}さんの傘ですか？」 "Is this Yamada-san's umbrella?" in the casual form:
これは山田さんの傘？ -- commonly used by both genders of all ages
これは山田さんの傘か？ -- fairly common but I think it's more used by males, can sound blunt/strong
(×これは山田さんの傘だ？ / ×これは山田さんの傘だか？)
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.
