This answer is specifically about examples 2a and 2b:
It is conventional in conversational English pair the phrase "ask [direct object]" with a preposition, such as "about":
I asked her about the kind of music that she liked.
However, this subtly changes the meaning of the sentence. To make the sentence sound more natural without changing its meaning, remove the direct object ("her").
I asked the kind of music that she liked.
The biggest exception is when a question word (who, what, when, where, why, how) is used immediately after "ask," as in example 2a:
I asked her what kind of music she liked.
That question word is a clue that what follows is not simply the topic of the question, but a restatement of the question itself. Example 2a could be rewritten as:
I asked her: "What kind of music do you like?"
Your original phrasing is more flexible, though, because it doesn't have to specify the exact words that were used.
You could also use the same trick that I mentioned before by removing the direct object:
I asked what kind of music she liked.
Removing the direct object might remove some context, but additional context will be required no matter what construction you use. Despite having a direct object, example 2a could mean
I asked (a woman) what kind of music (that same woman) liked.
I asked (a woman) what kind of music (a different woman) liked.