Learn Farsi: او در دانشگاه عربی می‌خواند.

The word عرب (Arab) refers to the people. By adding the suffix -ی (-i), it becomes the relational adjective عربی (Arabi), which literally means 'related to Arabs'. This is the standard way Farsi creates these adjectives.

The lesson is about adjectives, but here عربی is used to mean 'the Arabic language'. Does it act as a noun?

Yes! Many relational adjectives ending in -ی can be used as standalone nouns to refer to a language. So عربی can describe a noun (like Arabic food) or act as a noun itself (the Arabic language), just like in this sentence.

I learned that خواندن means 'to read'. Why does می‌خواند mean 'studies' here?

In Farsi, the verb خواندن (to read) is commonly used by itself to mean 'to study' an academic subject. While you can use the phrase درس خواندن for studying in general, just saying someone 'reads' a specific subject like عربی perfectly translates to 'studies' in English.

Why does the sentence use در (in) instead of a word for 'at' for the university?

Farsi doesn't make the same strict distinction between 'in' and 'at' as English does. The preposition در is used for being located in or at institutions, buildings, and cities, making در دانشگاه the natural way to say 'at university'.