In England, traditionally (for example in the 1600s, 1700s, and 1800s) students at the great British universities were expected to know Latin and Greek. The United States inherited this. Thus, up until the 1880s a knowledge of Latin and Greek was required to enter Harvard University. In this cultural context, "a Greek scholar" most likely means a university professor, whose native language is English, who has made a specialty of studying and teaching classical Greek. Because Latin and Greek were important university subjects, there would have been many such scholars. (At elite U.S. universities the Greek requirement began to be eliminated in the late 1800s, and Latin in the 1930s).

It depends on the context, but usually it would mean someone who studies Greek (probably ancient Greek). It could also mean a scholar from Greece, but if that is the case, the context would tell you that. It does NOT mean someone who studies Greece, the country. I don't think there is a simple phrase for that; maybe a Grecian Studies scholar?

Oh, the stress: if it is someone who studies the language, the phrasal stress would be on "Greek": a GREEK scholar. If it were a scholar from Greece, it would be a Greek SCHOLAR.