I just registered to say: yes, it definitely can be done. I only recently started playing Civ and occasionally browsing these forums to pick up tips, but as it happens I played exactly this game last week. BTS, Monarch, Epic, Earth 18 civs, Huayna Capac, all other settings standard. I ended up winning diplomacy by my vassals voting me into power in about 1870, but conquest would have been fairly easy at that point. I reckon I could have capitulated the remaining stragglers in at most another 30 years or so. The only reasonably big civ left was Catherine. If I hadn't given so much land back to vassals I could have done domination as well, but that would have been very boring.
I think the key is that a fully settled South America with cottages, watermills, levees and the right civics can do anything. So I ignored all wonders in favor of settling all of South America as quickly as possible. I settled 2W on the coast and went straight for sailing (after mining for the gold), and put my second city south by the deer. Then I quickly build a city right next to the mountains in South-Western Argentina to grab the copper on the other side of the mountains on the first border pop, and a massive food city on the Eastern coast of Argentina to pump out settlers and workers. You cannot delay copper any later, as you get massive numbers of barbarians in this region. I was able to whip my first axe on the exact turn the first barbarian axe arrived.
From there, just pump out massive numbers of workers, settlers, axes, archers and quechuas. Advance a line of barb-killers north across the width of the continent, and build cities behind them, while spawnbusting the Western and Southern coasts. Most of my cities didn't get anything beyond a terrace and a courthouse for quite some time (and a bunch of quechuas for happiness), until I'd filled the entire continent. When I met the old world I was slightly behind on tech, but catching up fast. I just missed out on Liberalism, but could possibly have gotten it if I had been content to get something without prerequisites from it instead of trying to get astronomy. But aiming directly for optics and astronomy is probably better anyway. After teching astronomy myself, I beelined to the techs required for the powerful civics set of Universal Suffrage, Emancipation and State Property, as well as steam power for levees. Once I got that up, I was massively out-teching the AI.
I started attacking in ~1780, with tanks, marines, battleships and fighter-loaded carriers. I attacked Asoka first, who was tech leader and a peace vassal of score and power leader Qin (with 4 vassals total), so I could get a few bomber squadrons in before taking on Qin's stack. Capitulating Qin took ages because he split off a colony in the pacific that wouldn't break off, giving him a warped power rating (I had to beat him down to a single city in Siberia), but after that everyone else fell quickly. By the end, I had two stacks going on the old world mainland, one in North-America, and one on ships in the Pacific, all modern armor and mechanized infantry with bomber support, while never facing anything more advanced than infantry, machine guns and artillery. The only leaders I didn't capitulate before my vassals voted me world leader were Catherine, Mansa and a couple of small European states.
So in short: colonize the continent upwards from Southern Argentina, take the barbarians very seriously, build all your cities on rivers, and once you have towns (growing quickly with emancipation), watermills and levees in free speech, universal suffrage and state property you will top the demographics in all categories. From there, any victory condition is available (well, maybe not culture - in my game the new world didn't get a religion and I didn't try for any of the early to mid-game wonders that boost culture). Judging by how easily my wars went, I reckon I could make this work on emperor fairly easily, and a better player could definitely make it work on immortal.
As for some of the specific issues raised in this thread: I personally didn't bother much with GP farming for a long time (just settle, settle, settle!), but my settler and worker pump on the Argentinian coast did fairly well as a GP farm later on, and I did one or two bulbs on the way to education (which should probably have been used on the way to astronomy instead). Early war is completely unneeded (no one can reach you and Monty is busy beating on Roosevelt) and you wouldn't have the production for it anyway before levees/watermills/universal suffrage. I also don't see why you'd bother with the great lighthouse: North America doesn't even count as overseas (I think) and Monty will most likely pressure you to close borders with Roosevelt, so trade income is very limited until astronomy. After astronomy your tech rate ramps up massively and you get to corporation very quickly. Or is the GLH already worth it for the internal trade? My estimation of trade so far is non-overseas = non-worthwhile, but I could be way off in that.
/ turn at size 3. It has got good food though, so whipping OF into the GLH is the way. Then probably double Astro bulb and GG.
will obviate HR.
plus some US 
economy?
