Definitely not at the same level or in the same fashion as their brief hegemony over Ancient Greece. By the time of roman ascendancy I believe that Sparta was reduced to a village of just a few hundred. Greece was not an important region after it's conquest by the ottomans and Sparta itself remains a fairly unimportant area.

No, not at all. After losing hegemony to Thebes at Leuctra (see Agrippa 911's answer) Sparta never again regained any real measure of power. When I was in college, my professor (JE Lendon) described them as becoming a sort of backwater "colonial Williamsburg," in which Romans came to see where the once-mighty Spartan warriors lived. Like Williamsburg, it became an important historical site that long outlived its place of power in the world. Visiting Sparta also reinforced Romans' sense of accomplishment that they had conquered Greece. I think the festival of Artemis Orthia, which had been a serious religious festival and competition for presitge among Spartan boys, became a sort of tourist spectical for Romans. I would highly recommend Lendon's book "Soldiers and Ghosts" which is a history of the military tactics that propelled different city-states to power througout the Greek and Roman period.