I don't exactly mean a culture bonus that starts in the medieval age literally but one that would really start working at its most powerful at that time (at perhaps approximately that 500 AD date you gave for the unification of France). For instance, Korea kicks in at the late medieval/early rennaissance when you get universities up, or Babylon, which begins working at writing, but really kicks in when you get universities. That's just an example, though.
Well, I didn't mean necessarily that French should completely eclipse its conquest capabilities, but that the current mixture of TWO unique units instead of one unit and a unique culture building doesn't really reflect its legacy as a cultural power (I'm thinking primarily of the art created during the Impressionism/Post-Impressionism era, which fits I think very well with great artists). Perhaps a mixture of culture and war makes the most sense somehow?
What I find off is that both UUs are gunpowder units, and for most of its period of military supremacy, France relied most heavily on superior cavalry, not gunpowder infantry. The Musketeer's fair enough for all that it owes more to Dumas than to history, but the Foreign Legion is a bizarre choice of UU.
Of course with a civ like France or China the developers are always going to have to make some arbitrary choices - these are countries with a very long continuous history (France is the oldest country in Europe), and without a single discrete 'golden age' that can be identified (unlike other long-lasting countries like Persia, or historically important countries that have only been resurrected in the modern era - such as Greece).
Maya is weak because the extra great people are not free. At first glance, it looks cool to get great people to pop automatically from the long count, but you are forced to take great people you don't necessarily want and the cost of future great people increases with each "free" one generated by the long count. In the long run, it is more of a unique penalty.
Great People you generate normally - or from Pisa or Porcelain Tower, come to that - aren't free either. Does the Babylonian GS now increase future GP points? There's no penalty to the Maya UA - you get exactly the same increase in GP cost you would from generating 5 GPs (Admirals and Generals don't increase the GP point count) normally, you just do so more quickly. And there's nothing weak about a civ that gets an Academy a couple of turns after Theology - look how favoured Babylon is for getting the same deal a couple of techs earlier, and without following that up with well-timed GEs or Great Prophets, or the science boost from Pyramids. The only limitation is that you have to select GPs that may not be optimal for your strategy in place of later GPs that might be better, but that's not a restriction that makes the Maya weak - it's one that stops them from being too powerful.
I took from the Swedes this time. I'm just not a fan of the UA. I don't like giving other civs a bonus in order to receive my own bonus.
Bear in mind that a 10% bonus on GP production is meaningless to an AI opponent - AIs never use specialists and don't know how to efficiently generate GPs, so they only tend to gain GPs through Wonders or through gradual accumulation of GP points from Wonders. Also, in a multiplayer game with allies, or single-player with reliable friends, you may well want to give the other civ a bonus that will give them an edge over your rivals without giving them an edge over you (since they only ever get 10%, while the bonus is cumulative for you).
Arabia 22
Aztecs 21
Babylon 22
Byzantium 16
Carthage 23
China 24
Egypt 6
England 20
Ethiopia 16
France 13
Greece 22
Inca 26
Iroquois 8
Japan 16
Korea 24
Maya 25
Mongolia 12
Netherlands 23
Ottoman Empire 17
Persia 17
Roman Empire 22
Russia 22
Siam 18
Songhai 1
Sweden 3
Sweden: As others have noted, this civ doesn't deserve to go this early. Nor does Songhai, but although they're a long-time favourite of mine, I have to admit that Sweden has a more interesting play style. I like diplo civs, and Sweden is, as others have noted, truly unique in the way it approaches this type of game.
Arabia: I'm a little wary of downvoting a civ I haven't played, but I do object in principle to ranking a civ highly based on a UB that only works because of bad AI programming. If the game is patched so that AIs either have less free gold or are less willing to spend it, Arabia's fortunes will go right down because, despite a good UU, its UA is not that impressive. A civ that's so dependent on programming quirks - and that is essentially useless in multiplayer - certainly doesn't deserve a place close to the top.