Learn Turkish: Çok zeki bir öğrenciydi. - She was a very intelligent student.

Yes! In this specific sentence, zeki is acting as a normal adjective modifying the noun öğrenci. But if you wanted to drop the noun entirely and just say 'She was a very intelligent one', you would attach the past tense suffix directly to the adjective, giving you the substantivized form Çok zekiydi.

Why is the word bir placed after the adjective instead of before it?

In English we say 'a very intelligent student', but in Turkish, the indefinite article bir must go directly before the noun, after any adjectives. The structure is almost always adjective + bir + noun, which literally reads as 'very intelligent a student'.

What exactly is the -y- doing in öğrenciydi?

It acts as a buffer letter. The past tense copula ending -di (was) actually comes from an older, separate word: idi. When attaching it to a noun that ends in a vowel, like öğrenci, that original i turns into a -y- to prevent two vowels from clashing.