First: Good job on counting the beakers! I just added up the turn numbers written on each tech, but since those numbers don't account for beaker overflow, and our output has increased, it looks really good! Problem is, we won't be able to get to legendary that soon. It means that building culture in Orleans is compulsory as soon as Orleans gets all its culture buildings. We also need our megacities at size ~22 (some bigger) to be able to starve enough great artists.
Check the tiles in Orelans. When building culture, the weaker hammer only tiles are usually worse than an artist specialist, since our culture multipliers are better than hammer multipliers there. If there is food though, then keep working it, we need to grow.
We need to move on with warring as well. The transportation time to the other continent is 6-7 turns at least, which is a lot with this timeframe. I wouldn't bother with too many infantry, they are slow. We need two of these for stack defense in the Babylonian war, but 4 will suffice for sure. After that I would stick to tanks, marines and some fighters/bombers + transports, carriers and battleships.
Which means we need to get combustion, industrialism and flight in this order.
Beakers into tech: there are two factors. One is the number of known civs that already has the tech. civ. (On this map, this gives 4% beaker increase for every known civ that already has the tech) The other is number of prereqs researched, and it is basically 20% for each prereq.
See more detailed explanation here:
http://forums.civfanatics.com/showthread.php?t=146163