What tech do you use to enter Medieval? and What is your favorite order of techs/builds for a Medieval
Most common for me would be Alpha or Math (trade for the one you didn't get) > Currency > CoL > CS > MC > Machinery > Eng. You can almost always trade for Constr and quite often MC. W/out fishing you can always make a few bulbs.
To enter Renaissance? and Renaissance warmonger?
Once again, Alpha or Math (trade for the other you didn't 'get) > Currency > Aesth > Lit > Music > Philo (bulb) > Nationalism > Paper > Edu (bulb/s) > Compass (trade bait + allows you to bulb Lib) > Lib (bulb + free tech MT) > GP > Start Whipping Cuirs. Techs not mentioned like Calendar, Sailing, Med, MC, etc........are all traded for.
What is your ideal number of cities to maintain at each of these stages?
Every game will be different depending on your land and neighbors. Here are some nifty little tips/facts that most people overlook: Take a look at the attachment below. 1 city. 2 workers. 2 pre chopped forest. 1 farmed FP or any other +2F improved tile. 1 Missionary. If you don't have a +2F source already farmed bring extra workers to finish the food in 1-2 turns.
Spread the Religion (you already in ORG Rel), chop 2 forest, improve the other food resources. Ideally you want 2 improved tiles that offer +4F. This could like like 2 improved food plains...or 1 improved wet wheat + 1 grassland farm, etc. This allows you to grow from 2 pop to 3 pop in 2t. Likewise, at size 3 all you need to do is work a 3rd tile with +1F which is easy with irrigation. Just farm a grassland. This allows you to grow from 3 pop to 4 pop in 2ts. Working a 4th tile with +1F likewise allows you to grow from 4 to 5pop in 2ts.
After you've chopped your granary you improve the remaining food (4tiles is enough) and at size 6 whip the forge. The very next turn you have your first Unit Qued up (turn 15), and at turn 16 you can whip. Keep whipping and in 30t from founding this late city you'll have 5 Macemen. If you found this city at 1AD you'll have your first unit finished at 400 AD and by 680 AD you'd have just whipped your 5th Macemen.
If you settled 3 cities with this set-up at 1AD that's 15 Mace at 680 AD. And we're talking just those 3 cities. Similarly, if you were going to Cuirs route you would do the same thing. The difference is you'd get more time until you reached the Cuir tech so you'd also be able to whip out a barracks. So with whipping Cuirs, any city you found at 1AD will take 37t to get a granary, forge, barracks, and whip 4 Cuirs (start whipping around the 500-550 range). If you won't reach Cuirs til after that then adjust the 37t to coincide with your projected MT/GP completion date.
Exp leaders only need 1 Tree to get the ball rolling while IND leaders will have the additional time to whip a barracks in the first scenario and whip a stable in the second.......or keep the same infrastructure but start cities a little later. So keeping some space for some seemingly horrible cities who need nothing more than 4 modest tiles to be worked will pay off huge dividends via the whip later on.
So the point of all this rambling is don't be concerned with an exact amount of cities. If you've got room to expand and good sources of commerce then good! But if you don't, don't stress. You can still keep your empire smaller in a pinch and with some good planning get 2-3 filler cities placed down at 1AD and do some serious damage with some unimpressive cities.
What troops do you favor in Medieval/Renaissance?
Medieval: Mostly Mace/Treb with some Phant/Pike mixed in for anti Horse units. Renaissance: Cuirs will always be the first choice, followed by Cannons supported by anything. Plus this keeps us on the bottom of the tech path which is better imo, seeing how you're probably continuing a war from the Engineering path anyways.