In english we don't normally say "a" heart disease. That implies more than one. We just say "heart disease".
We definitely do say 'a heart disease' because there are more than one.
It seems like in this phrase that "heart" is being used as an adjective. But I guess seeing the declension, it's literally saying "A disease of the heart". So heart is still used as a noun, right? You just decline it using genitive?
Yes, it's just that different grammatical constructions seem more natural in each language. So in Polish it's rather "a disease of the heart". Genitive, yes.