As I said I played them all on viceroy in vanalla so I'm not sure how these stragegies are compared to other difficulties.
The hardest ones I found were the ones you said Persia, Incas and Mongols
Persia: Go after India first, you starting military should be enough. I usually put my extra settler on the horses to the north because of the barbarian activity. My beginning archers go after the barbarians to the left. Workers should connect your cites while you cities should build horses. After India is secure take whats left of the barbarians.
At this point I have my capital build wonders while my army goes and takes Greece if they are weak or Egypt. If you take Greece, take Egypt next. Keep checking f8 stop if you want when you hit about 7%. While you are doing this research to music and build a settler. Build a city in the middle of africa and use the great artist there, this should give about 8%. Other then that run specialists in Delhi for two prophets. When your land is secure finish what wonders are left.
Incas: The hardest civ to be. Research bronze working and turn off research and get slavery. Slow research to sailing and then any tech that will get you to guilds. For cities I took the 4 settlers, founded the capital, one city to the south. Razed the barbarian city and put a city one square south of it on the desert and the fourth to the north of it near the gold. Have all cities building barracks the north city builds walls first. Have workers hook up copper. After barracks build catapults, axes and spears, whip whenever you can. Put all these units in the north city. Use the barbarian axes to upgrade an axe to level 4 and send him out for mersenary duty. Run specialists in a city, I like to run them in the city west of capital and save those great merciants.
When your units start to cost you money switch cities to wealth and move a warrior in the north to the gulf of mexico. This is where you need some luck. When they find you they will attack first, hope that its in the north city. Also hope that they don't have gunpowder or horses yet. The time I won England had maces, pikes and catapults. After they attack you can attack back, clear them all. After you get sailing build a boat and put any GM on them and send them to the aztecs. Trade with the aztecs any way you can get techs that put you closer to guilds. After the event and you secure your cities start building settlers.
Turn back on research and go the rest of the way to guilds, trade as much as you can. Send you military out to cover the coast looking for anyone who is going to land. If they do, declare and kill their units. This is where you need some luck again. In my game my largest stack of axes had France land right next to them. Get peace as soon as you can. After you get guilds you can turn off research again or go for banking, it depends on the date. As you build more cities your gpt should go up. When it doesn't stop and start building units again. Get grocers whenever you get guilds and banks if you get banking. You should get a second great merciant by now which should also go to the aztecs. Once you hit 5000 gold make sure you settle the entire contient and the third condiction should be easy.
Mongols: Declare on China in the beginning. The army in the beginning should be able to do it. Have all cities build settlers and have them settle all of siberia. Use Beijeng get you a great artist. Use them to expand your boarders to where you can't settle. There should be enough space in siberia and eastern Russia. Doesn't matter how useless the cities are. Didn't even put anyone in the cities until I was near 12%. After I have my 12% then I built some units and launched an attack on India.
Hope this helps, Incas by far took the longest and I didn't like how much luck it involves, still think they need one commerse on the mountains because you don't want to use the mountains if you are going for 5000 gold. Good luck