Learn Croatian: Slušajući glazbu rješavam problem. - While listening to music, I solve a problem.

It is the present verbal adverb of the verb slušati (to listen). To make it, you take the "they" form of the present tense, which is slušaju (they listen), and simply add -ći to the end.

Could I use slušajući if I was listening to music while someone else solved the problem?

No, you can only use this -ći form when both actions are done by the exact same subject. Because the main verb is rješavam (I solve), the person listening to music must also be "I". If the subjects were different, you would have to use dok (while), like Dok ja slušam glazbu, on rješava problem.

Why is the word glazbu in the accusative case, even though it follows an adverb rather than a normal conjugated verb?

Even though slušajući functions as an adverb here, it is still built from the verb slušati, which takes a direct object in the accusative case. Verbal adverbs keep the same case requirements as their original verbs, so glazba becomes glazbu.

Does the main verb rješavam have to be imperfective because of the present verbal adverb?

No, the main verb can be perfective or imperfective. For example, you could say Slušajući glazbu riješio sam problem (While listening to music, I solved the problem). However, the verbal adverb itself (slušajući) must always be formed from an imperfective verb, because it describes an ongoing background action.