You are correct that "what" refers to a thing, whereas "how" refers to the qualities of a thing. However, "How does it sound like?" is not correct. You could use either "How does it sound?" or "What does it sound like?" (both are correct, in this context the meaning is the same, but they aren't generally interchangeable).
To see the difference, look at how you would use 'sound like' verses 'sound' not in a question: E.g., if someone has a foreign accent, you could say either "He sounds foreign" or "He sounds like a foreigner". In "X sounds Y", Y is some trait that you're attributing to X based on the quality of the sound. In "X sounds like Y", Y is some noun that your saying X sounds similar to. Thus, "what" is correct with "sounds like" because "What does X sound like?" will have the answer "X sounds like Y" where Y is some thing with a similar sound to X. "How" is correct with "sound" because "How does X sound?" will have the answer "X sounds Y" where Y is some property that X might have, based on the sound of X.
There's also another usage of "Sounds like", which is "X sounds like X is Y", for example "He sounds like he is a foreigner". This has the same meaning as "X sounds Y". But you still should ask "What does X sound like?" instead of "How does X sound like?" because the answer would be "X is Y" describing a state of being, not a quality of the sound.
In contexts other than this particular one, there are further differences in nuance between "X sounds Y" and "X sounds like Y".
When you say "X sounds Y", you are saying that it's plausible that X is in fact Y based on the sound. Returning to my previous example, "He sounds foreign" is implying that your best guess, based on the sound, is that he is in fact foreign. On the other hand, "X sounds like Y" usually (but not always) has the connotation that X is not actually Y, but only sounds similar to how Y sounds. If I say "He sounds like a foreigner" in the context of someone having an accent, most likely I would be complimenting his ability to mimic a foreign accent, not saying that he probably is in fact a foreigner. For this reason, don't use "X sounds like Y" if you're not sure whether or not X is in fact Y.
You should also note "How does X sound?" is most commonly used to ask for someone's opinion about a suggestion, for example, if you're planning a party, you could say "Let's start at 6:00, how does that sound?" or "How does it sound if we start at 6:00?" to ask people's opinion about 6:00 as a starting time.