The cold has settled in my chest and I have a cough and headache.

Literal

Cold [subject-が] chest [target-に] came-and, cough [and-と] headache [subject-が] do.

胸にきて means 'has come to the chest' — the cold has moved to the bronchial area. 咳と頭痛がする uses する for experiencing symptoms. と lists both symptoms.