I think you've become confused here. It is much easier than that: words like что and как do not experience any agreement because they are not modifiers.
You can consider each question word as a replacement of a certain type word or a phrase. You will get such phrase as an answer to a question:
Что is a dummy noun (and so is кто). That's why it does not have to agree with the noun—it modifies no noun and could care less. When you use что in a question, you expect that a noun is the shortest answer (naturally, a question like "What's going on?" usually elicits a more elaborate answer):
- Что ты ешь? = What are you eating?
- О чём ты думаешь? = What are you thinking about?
Как is a dummy adverb, a way of performing an action; in Russian adverbs do not have any forms at all (apart from comparative degree for adverbs like "fast"):
- Как ты открыл дверь? = How did you open the door?
Какой is a dummy adjectival modifier. When you use it, you expect an adjective or a description as an answer:
- Какой свитер лучше? = Which(What) sweater is better?
Чей (whose) is a question word for possessive modifier:
- Чья это кошка? = Whose cat is it? (note the unchanging это)
Где replaces an adverb of place or a phrase describing a place:
- Где ты живёшь? = Where do you live?
Similarly, куда and откуда replace and adverb (a phrase) that means a direction TO some place or a direction FROM some place:
- Откуда вы? = Where are you from?
- Куда она переехала? = Where did she move to?
Зачем and почему describe a purpose / a cause (you can easily imagine what type of phrase they stand for).
It does but usually under a limited number of contexts. We do not have any sentences like that in the course. The most prominent is "Который час?" for asking the time.
Sometimes we use it to ask for the ordinal number, there are also instances of "Which one (exactly)?" , though Russian National Corpus has, maybe, five such sentences in the last 100 years.
- in the latter role, it is most useful when used as an unexpected reply, (e.g., "How is your boyfriend?" "Which one?")
The English course for Russian speakers uses который extensively to sifferentiate between "what" and "which". Truth be told, this is understandable by a native but not how people actually speak.
I was marked "correct" by Duo for that word-order. (Y вас есть kакая музыка ?)
Shady_arc's comments gives me pause to consider the "rule" that the more important information in a sentence comes at the end. It is fairly certain that the the person being asked does have music, and that the focus is on the kind of music, so the "importance of information rule" would suggest that какая музыка should be placed at the end of the sentence.
I recognize that idiom trumps such "rules", so I'm not arguing for this placement, but more being curious as to why it doesn't apply here. The more I learn about usage and "rules", the more I change the order of words in Russian to be more natural, so just because something seems odd from an English-speaking standpoint has little relevance to how grammar works in Russian, even though there are a substantial number of logical parallels.
"What do you have?" is OK. "What kind of music do you have?" is OK. "What music do you have?" is bad English grammar, not logical and requires assumptions on the part of the listening person, assumptions that may not match the speaker's thoughts. "Which do you have?" is not OK. "Which kind of music do you have?" is OK. "Which music do you have?" is OK.
A trick I do when something is confusing or odd about an exercise: before writing my translation, I high-light and copy the exercise sentence, open a new tab or window, go to Duo's discussion area, and paste the exercise sentence into the search box. I find the discussion of the exercise, and read about it so I know what I'm doing when I answer the exercise.
I also sometimes copy the exercise into an on-line translator before filling in the answer, in order to see what they say. Or search on phrases like "как vs. какий". [In doing comparison searches, using "vs." is very common, so it's quite useful.] Or searches include "declension table for какий" and other words like that.
In a school setting, that would be "cheating", maybe, but this isn't school, everything is "open-book", and I'm a strong believer that getting things right the first time is much more conducive to learning that getting something wrong and having it corrected - because the first thing you "learn" is the wrong way, and that's stupid. Why spend valuable minutes, which add up to hours, being frustrated and wandering in the dark. It all gets repeated so many times in Duo that getting it right the first time is a real aid in learning.
Which music is a better translation. You could ask What kind of music do you have but asking what music is just plain wrong. People will use "what" in everyday language but people mangle the English language in many ways and that does not convert bad grammar into acceptable grammar.