is it possible to say: moje levé oko bolí? or would it be unnatural?
It's understandable, but it's obviously a direct translation from Germanic languages and it's not used.
Bolí mě oko, bolí mě břicho, bolí mě hlava, bolí mě záda... celé tělo mě bolí. It's always literally "something pains/hurts me", not "my something hurts".
Why is mě required here and why can't it be the first word in the sentence ?