Learn Polish: Opanowała nie tylko matematykę, ale i sztukę. - She has mastered not only mathematics, but also art.

Yes, ale i, ale też, and ale także are completely interchangeable here. They all form the 'but also' part of the nie tylko ..., ale ... structure taught in this lesson.

Is the i strictly necessary? What if I just said ale sztukę?

It is highly recommended to include it. Without an additive word like i, też, or także, the sentence sounds unfinished to a native ear, because nie tylko sets up a strong expectation for an 'also' in the second clause.

Why do both matematykę and sztukę end in -ę?

They are direct objects of the verb opanowała, which requires the accusative case. The conjunction nie tylko ..., ale i ... smoothly links two items of equal grammatical weight, so both feminine nouns change their dictionary -a ending to -ę.

Why use the perfective verb opanowała here instead of an imperfective equivalent?

The perfective aspect shows a completed action with a definitive result. It highlights that she has fully achieved mastery over both subjects, rather than just being in the process of learning them.