Learn Hardcore German: Ich lese ein Buch und höre gleichzeitig Musik. - I read a book and listen to music at the same time.

Because Buch is a neuter noun (das Buch). In the accusative case (direct object) a neuter noun keeps the same article ein in both nominative and accusative. By contrast, masculine nouns change ein → einen in the accusative.

Why is there no article before Musik?

Musik is an uncountable (mass) noun in German, so you normally omit the article when speaking in general. You say Musik hören, not die Musik hören, unless you refer to a very specific piece or playlist (e.g. die Musik, die gerade im Radio läuft).

Why is the verb in simple present? In English we’d say “I’m reading” and “I’m listening.”

German uses the simple present (Präsens) both for habitual actions and for ongoing actions. The English present continuous (“I’m reading”) is usually just Präsens in German.
If you want to stress the ongoing nature, you can add gerade:

Ich lese gerade ein Buch.

What does gleichzeitig mean, and can I use zur gleichen Zeit or zugleich instead?