Learn Turkish: Öksürüğüm dün gece başladı.

The base word for 'cough' is öksürük. When we add the vowel suffix -üm to say 'my cough', the final k softens into a ğ. This happens often in Turkish when a word ending in k gets a suffix starting with a vowel.

Does dün gece literally mean 'yesterday night'?

Yes! To say 'last night' in Turkish, you simply combine dün (yesterday) and gece (night). You don't use a word for 'last' here.

Why doesn't the verb başladı have a personal ending, since the sentence is about 'my' cough?

Even though it is 'my' cough, the grammatical subject of the sentence is the cough itself ('it'). Because it is a third-person subject, the verb başladı (started) only needs the past tense suffix -dı, with no extra personal ending.

I have seen the verb öksürmek (to cough). How is it related to öksürük?

They are directly related! öksürmek is the action of coughing, and adding the suffix -ük turns it into the noun öksürük (a cough).