Learn Russian: Моя жена увлекается русской музыкой.

The verb увлекаться (to be into / to be passionate about) requires its object to be in the bare Instrumental case. Since the Instrumental case is often used to show the means by which something is done, you can think of it literally as the wife being fascinated 'by means of' Russian music.

What are the specific feminine Instrumental endings used in this sentence?

For feminine nouns ending in -а like музыка, the Instrumental ending changes to -ой, becoming музыкой. The matching feminine adjective русская also takes the -ой ending in the Instrumental case, becoming русской.

Does увлекается have a non-reflexive form?

Yes! The base verb увлекать means 'to fascinate' or 'to captivate' someone else. Adding the reflexive ending -ся turns the action back on the subject, changing the meaning to 'to be fascinated by' or 'to be into' something.

Could I just use the verb любить (to love) here instead?

You could, but the grammar would change! Любить takes the direct object in the Accusative case, so you would say Моя жена любит русскую музыку. Using увлекается with the Instrumental case gives the specific nuance that it is an active interest or hobby rather than just something she enjoys.