My ideas:
(In terms of the issue of foreign domination, I've followed the rule that if the foreign 'invaders' didn't completely displace the populace, and if they ruled from the civ in question rather than the invader's homeland then they count as a successor state, and thus the same civ.
Also, where there's more than one possible name in the same period, I've gone for the one which lasted longer unless the other particular stands out as important.
I've also tried to match up leaders where possible.)
Egypt
Ancient - Kingdom of Egypt (Hatshepsut/Ramesses II)
Classical - Ptolemaic Empire
Medieval - Ayyubid (Saladin really, after all he wasn't Arabic)
Renaissance - Mamluk Sultanate
Industrial - Viceroyalty of Egypt
Modern - Republic of Egypt
India (I've concentrated on the north, since that's where you start in RFC)
Ancient - Indus Valley
Classical - Maurya Empire (Asoka)
Medieval - Delhi Sultanate
Renaissance - Mughal Empire
Industrial - Indian Empire
Modern - Republic of India (Ghandi)
China
(about Dynasties: I don't think there's anything wrong with using dynasty names. After all China as we know it was not always united, but slowly conquered by successive, distinct kingdoms which we call dynasties. But the comparison to European Royal Dynasties is false, not least because their names weren't always related to the name of the emperors)
Ancient - Zhou Empire
Classical - Han Empire/Han Empire (Qin Shi Huang)
Medieval - Yuan Empire (Kublai Khan)
Renaissance - Empire of the Great Ming
Industrial - Empire of the Great Qing
Modern - People's Republic of China (Mao Zedong)
Babylonia
Ancient - Babylonian Empire
Classical - Babylonian Empire
Medieval - Iraq
Renaissance - Iraq
Industrial - Iraq
Modern - Republic of Iraq
Greece
Ancient - Delian League (Pericles)
Classical - Macedonian Empire (Alexander)
Medieval - Byzantine Empire (Justinian I)
Renaissance - Byzantine Empire
Industrial - Kingdom of Greece
Modern - Hellenic Republic
Carthage
Ancient - Phoenician Empire
Classical - Carthaginian Empire (Hannibal)
Medieval - Exarchate of Africa
Renaissance - Tunisia
Industrial - Tunisia
Modern - Tunisian Republic
Persia
Ancient - Achaemenid Empire
Classical - Parthian Empire
Medieval - Sassanid Empire
Renaissance - Safavid Empire
Industrial - Iranian Empire (representing the constitutional monarchy)
Modern - Republic of Iranian (not Islam... because what if its not?)
Rome
Ancient - Senate and People of Rome (Julius Caesar)
Classical - Roman Empire (Augustus)
Medieval - Kingdom of Italy (there's an alt history aspect here, since I can't think of any way to represent all the city-states, and it did at least exist as a title for the HREs)
Renaissance - Papal States (solely because it was centred around Rome)
Industrial - Kingdom of Italy (ie after the Risorgimento)
Modern - Italian Republic
Japan
Ancient - Japanese
Classical - Japanese(afaik Japan was not unified until 600+ under the Emperor, and has stayed under the one dynasty since. Remarkable really. But before that there were only assorted tribes, no states to speak of. Why does it spawn so early??)
Medieval - Japanese Empire
Renaissance - Japanese Empire (Tokugawa)
Industrial - Japanese Empire
Modern - Japanese Empire
Ethiopia
Ancient - Kingdom of D'mt
Classical - Kingdom of Aksum
Medieval - Agaw Empire (ie Zagwe dynasty)
Renaissance - Ethiopian Empire (ie Solomonic dynasty, Zara Yaqob)
Industrial - Ethiopian Empire
Modern - Republic of Ethiopia
Maya
Ancient - Maya (because they were really just an ethnic group rather than any sort of state)
Classical - Maya
Medieval - Maya (Pacal II)
Renaissance - Maya
Industrial - Yucatan (my rationale: the Inca are usually gone by this time, so any resurgence will have the right name. Same goes for the Aztec and Inca)
Modern - Yucatan
Vikings
Ancient - Vikings
Classical - Vikings
Medieval - Vikings (Ragnar)
Renaissance - Kalmar Union
Industrial - Kingdom of Sweden-Norway
Modern - Kingdom of Scandanavia (a kind of hypothetical constitutional monarchy, if the Scandinavism movement of the 19th century had been successful like over nationalalist movements)
Arabia (very hard this one... since the term is so contested to start with and how after the expansion of Islam there were seperate Arabized states in Egypt, Persia and India)
Ancient - Arabs
Classical - Arabs
Medieval - Umayyad Caliphate (Saladin... I suppose)
Renaissance - Abbasid Caliphate
Industrial - Kingdom of Saudi Arabia
Modern - Kingdom of Saudi Arabia
Khmer
Ancient - Khmer
Classical - Khmer
Medieval - Khmer Empire (Suryavarman)
Renaissance - Khmer Kingdom
Industrial - Khmer Kingdom
Modern - People's Republic of Kampuchea
(with the European civs, I haven't generally used "Empire" because I think its too limiting. And they weren't like Rome or China, they didn't have an Empire. They were just Kingdoms or Republics with lots of overseas protectorates and colonies)
Spain
Ancient - Iberia
Classical - Iberia
Medieval - Visigothic Kingdom
Renaissance - Kingdom of Spain (Isabella)
Industrial - Kingdom of Spain
Modern - Kingdom of Spain
France
Ancient - Gaul
Classical - Gaul
Medieval - Frankish Empire
Renaissance - Kingdom of France (Louis XIV)
Industrial - French Empire (Napoleon)
Modern - French Republic (De Gaulle)
England
Ancient - Britannia
Classical - Britannia
Medieval - Kingdom of England
Renaissance - Kingdom of Great Britain (Elizabeth)
Industrial - United Kingdom (Victoria)
Modern - United Kingdom (Churchill)
Germany
Ancient - Germania
Classical - Germania
Medieval - Holy Roman Empire
Renaissance - Prussian Empire (Frederick)
Industrial - German Empire (Bismarck)
Modern - Federal Republic of Germany
Russia
Ancient - Scythia
Classical - Scythia
Medieval - Grand Duchy of Moscow
Renaissance - Tsardom of Russia (Peter)
Industrial - Russian Empire (Catherine)
Modern - Union of Soviet Socialist Republics (Stalin)
Netherlands
Ancient -
Classical -
Medieval - Duchy of Burgundy
Renaissance - Dutch Republic (Willem van Oranje)
Industrial - Kingdom of the Netherlands
Modern - Kingdom of the Netherlands
Mali
Ancient - Mali
Classical - Mali
Medieval - Ghana Empire
Renaissance - Mali Empire (Mansu Musa)
Industrial - Bambara Empire
Modern - Republic of Mali
Portugal
Ancient -
Classical -
Medieval - Kingdom of Portugal
Renaissance - Kingdom of Portugal (Joao II)
Industrial - Kingdom of Portugal
Modern - Portugese Republic
Inca
Ancient - Inca
Classical - Inca
Medieval - Inca Empire
Renaissance - Inca Empire (Huayna Capac)
Industrial - Viceroyalty of Peru (my rationale: the Inca are usually gone by this time, so any resurgence will have the right name. Same goes for the Aztec)
Modern - Republic of Peru
Mongolia
Ancient - Mongols Khanate
Classical - Mongol Khanate
Medieval - Mongol Khanate (Genghis Khan)
Renaissance - Mongol Khanate (Kublai Khan)
Industrial - Mongol Khanate
Modern - Mongolian People's Republic
Aztecs
Ancient - Aztec Empire
Classical - Aztec Empire
Medieval - Aztec Empire
Renaissance - Aztec Empire
Industrial - Mexico
Modern - Mexico
Turkey
Ancient - Turks
Classical - Turks
Medieval - Seljuk Empire
Renaissance - Ottoman Empire (Mehmed II)
Industrial - Ottoman Empire
Modern - Republic of Turkey
America
Ancient - United States of America
Classical - United States of America
Medieval - United States of America
Renaissance - United States of America (Washington)
Industrial - United States of America (Lincoln)
Modern - United States of America (Roosevelt)
It would be cool if city names could change along with the civ names, where appropriate.