Catholic URV as Spain achieved on third try. Once I figured out the basic strategy it was quite easy, probably could have achieved a hundred years earlier if I was more aggressive militarily and knew some of the strats I adopted throughout the game.
0 - 8th place in history for uniting all mankind (except the Russians and Swedes) under Catholicism? Seems rather cruel.
1 - Pope and 12 Great Prophets are easy. 50% territory - not population - is the tough part. Probably better achieved by spamming random Africa and American settlements, but oh well.
2 - Who cares if the Germans take Paris. They're Catholic - it counts to the goal!
3 - Main thing stopping me was a giga-Tamils that controlled everything from Delhi to Sri Lanka. When they collapsed it was easy to just conquer all India in thirty turns.
4 - I decided early on there was little benefit actually vassalising France/Portugal. Portugal can colonise on its own even without Lisbon (which I sadly had to return in a Congress), Britain will colonise France's territory. I much preferred free tiles for my core and Paris as a production city to spam missionaries. By conquering Moors early I went on a speedrun in Europe, Moors-France-Portugal-Italy-England-Morocco by around 1500. Marrakesh is a good city too.
5 - Africa. Morocco vassalised briefly, but they collapsed quick. Mali was also a vassal, also collapsed. West African territory wasn't worth the effort and stability, maybe better than India though?
6 - Middle East. Vassal Turkey controlled Greece and Egypt, and for some very lucky reason AI founded Rafah (Suez) - might not have been so easy if it didn't. Turkey lost Greece and Egypt, but I'd put the effort in to convert them, so they spawned Catholic. But for a few hundred years control of Suez-Mecca-Sanaa was vital to get my ships from Cordoba to Asia.
7 - Asia. Conquered Indonesia, then Japan, then Korea, then Southeast Asia, then Tibet, then China. Got lucky because Ming spawned with ~6 arquebuisers per city, but they collapsed, leaving me independents and barbs to gobble up. At one point all of China and Japan were Catholic but Japan went secular for a while.
8 - Catholicism actually sat at around 50% influence most of the game. Probably peaked before Japan and Britain went secular.
9 - Stability was at Solid most of the game from Medieval Civics plus stable vassals. Expansion stability only collapsed after I conquered India and China, when I knew I could sprint for the goal.
The tricks learned:
* Early conquest. Took Barcelona with 1 swordsman lost. Then took Marseille and razed it with 1 swordsman lost before France even spawned. Went Conquest, built a small army without much siege and a few skirmishers, lots of Horse Archers to kill Moors in the field, took Cordoba by 1000. Built a little bigger army then took out France fairly easily. By the time you kill France, you have enough hammers that Portugal and Italy are light work. I wanted Holy Rome too but Turks conquered them, no luck. Built some cogs and vassalised England. Most conquering was done by bombards and heavy swordsmen, added Tercios later.
* Razed Marseille, Bordeaux and later the Italian Nizza in order to maximise tiles for core. Suffered expansion stability but who cares at that stage in the game. Founded Pamplona in the Renaissance to work French tiles for extra core pop. Turned off growth in all non-core and non-historic cities (later Paris and Bangkok growth turned on).
* No colonisation. Didn't build a single American colony. Catholic Portugal and England did all the work for me. Swedes and Poles took Aztec and Inca, both collapsed but fortunately spread Catholicism there for me. Make sure to grab circumnavigation for +3 movement on all vessels.
* Core expands in Renaissance. I rushed Firearms - no need for Exploration rush, Tercios were more important.
* Conquered Eastwards. Upgraded Cog fleet to Galleons, took Indonesia, then Japan, then Korea. Vassalising was very easy. Build Missionaries in sets of three, convert conquered cities, gift to vassals, diplo ask them to convert. Do this rigorously as many vassals have Theocracy so you must spread religion first.
* Run Theocracy. Buddhism, Islam and other pesky religions will naturally disappear. After you convert vassals, convert them to Theocracy. They will passively remove hostile religions, meaning when Civs respawn only Catholicism should be present.
* Barely touched Civics. Monarchy-Vassalage(later Constitution)-Manorialism-Regulated Trade-Theocracy-Tributaries the whole game. +20 Civic Stability until Industrial era.
* Economy dies at a certain point from excessive conquest. Realised I should have just built gold to spare cities (spare cities = less occupation timer = quicker to gift to vassals to convert them), (more gold = run culture = more land).
* Keep gold spare or trade for gold to ensure you can convert England/Japan to Theocracy if they go Secular. This strategy probably would have failed by the time other civs went secular, luckily I beat the clock (even at my own slow pace).
* Wonders aren't really needed. In fact I probably could have run wealth in most cities. Only cities that really mattered were Madrid, Paris and Marrakech for hammers (later picked up Xian and Bangkok for some Asian ships/units), and then Santiago/Cordoba/Barcelona/Pamplona for population and some commerce.
* Scientific Method was probably stupid since I grabbed it before building monasteries in Asia, would have been even quicker if I got a monastery in Bangkok.
* If you're vigilant with civics you could probably keep Moors/Mali/Tibet/others alive longer, but I wasn't too bothered.