I take it: sick and tired doesn't have the meaning of: fed up. E.g.: I am sick and tired of your whining.
It doesn't in Turkish. If only! :D
Does "Indian" in the translation mean South Indian, from India (a Hindu)?
From India, in Turkish the American indians are called yerli which can be translated as indigenous :)
What's the problem with "An Indian man is sick and tired"? Is that an accusative case?
Your sentence means "Hintli bir adam hasta ve yorgun". Here the sentence in question is about a specific Indian man rather than a random Indian man, therefore you have to use "the".
In addition to what Ilkei said, essentially in Turkish, a noun preceded by an adjective is assumed to be definite unless it is preceded by "bir" (after the adjective).