How did early civilizations deal with democracy? What were its major limitations? Is democracy compatible with running an empire?

Democracy was a very rare phenomenon in antiquity. Most of the great empires, such as the Egyptian, Mesopotamian, and Persian ones, were actually monarchies, often theocratic in nature. The two ancient civilizations that experimented with democratic governments had only limited success with the experiments.
The first major democracy, and one that has very much influenced the subsequent growth of this form of political organization was that of Athens. Strikingly, it was a direct democracy in which all citizens could participate by speaking in the Assembly and voting directly on laws. Many of the important positions in Athens were selected by lot to prevent corruption. What made this workable is that there were fewer than 50,000 citizens in Athens, and normal attendance at the Assembly was probably around 10,000. Membership in the Assembly was restricted to males over the age of 18 with Athenian parents or grandparents. Women, slaves, and metics (resident aliens) who constituted the actual majority of the population, could not vote.
The so-called Athenian Empire began as the Delian League, an alliance among Greek city-states against Persia's efforts to conquer Greece. After the defeat of the Persians, Athens, although internally a democracy, behaved increasingly tyrannically in the Delian League, looting its treasury to rebuild the Acropolis and treating other members as subordinate, behavior that soon led to the Peloponnesian Wars. Thucydides argues that democracy led somewhat inevitably to tyrannical behavior, as demagogues like Clean would inflame nativist and oppressive tendencies to maintain their power and popularity. As the "Athenian Empire" lasted only 50 years, we can say that the Athenian democracy did not display much competency in imperial rule.
Rome also began as a small Republic, but its administrative mechanisms were not really adequate to the task of running the empire it almost accidentally acquired after the Punic Wars. Rather than extending a republican form of government to conquered territories, it tended to send provincial governors who acted as absolute rulers. In certain ways, the necessity of administering an empire led to the collapse of the Republic and the beginning of monarchial rule.