Learn Hardcore German: Gestern hatte ich schon Kopfschmerzen, bevor die Klassenarbeit begann. - Yesterday I had already had a headache before the test started.

Test: very general, can be short; “quiz/test”
Prüfung: “exam,” often bigger and more formal (final exams, entrance exams)
Klassenarbeit: specifically a school class’s written test (e.g. a math test in 7th grade).

Breakdown of Gestern hatte ich schon Kopfschmerzen, bevor die Klassenarbeit begann.

Why does the sentence start with Gestern hatte ich … instead of Ich hatte gestern …? Are both correct?

Both are correct. German has flexible word order for elements like time, manner, and place.

Gestern hatte ich schon Kopfschmerzen, …
– Emphasis is a bit more on “yesterday” as the time frame.
Ich hatte gestern schon Kopfschmerzen, …
– Emphasis is a bit more on “I had” (the state of me having a headache), with gestern just giving extra information.

In both cases, German keeps the verb in second position in the main clause:

So the choice is mostly about style and emphasis, not grammar correctness.

Why is it hatte and begann instead of something with haben like habe gehabt or hat begonnen?