Thoughts on the civs I've played at least once. For reference, I play on the medium difficulty settings - I'm just not that good - and am attempting to adjust my impressions of strength for the difficulty.
Iroquois - The UA is sort of marginal-feeling, even if you're in a heavily forested area. (And you typically will be because the game attempts to start you in one.) It's probably actually pretty decent, since it saves you a few gold a turn, but the movement aspect of it is pretty unimpressive-feeling. The UU is only very marginally better than what it replaces. The Longhouse is decent, allowing you to get 1/2/2 out of forest tiles (with a trading post). Overall the package seems pretty weak, but I like that the bonuses are focused in a way that encourages approaching the game a little differently.
India - Hot dang. The reduction in unhappiness from size is so much larger than the increase from number of cities that I was able to basically blow off happiness improvements and was still regularly getting happiness golden ages. You don't have to twist your normal play mode at all to get a huge benefit from the UA, unless your normal play mode is "tons of size two cities" or something. A "normal" empire with a fair number of cities of different sizes still is massively more happy as India. The UB is pretty nice (I was only on Prince this game, so defending myself was less of a priority, but post-flight the buildings are all upside), although I never used the unique unit.
China - Other people have said this, but the great general bonus is just silly powerful. Even if you only engage in a moderate amount of conflict, the extra number of them you get is quite noticeable, and as other people have said, the extra power makes it as though all your units are UU. Paper maker is an amazing unique building as well, taking something you already want to build and making it give you three gold instead of costing one. Cho-Ko-Nu are okay; one thing I appreciated about them that I hadn't thought about was the ability of a single Cho-Ko-Nu to "mop up" multiple heavily injured enemy units in a single turn. I believe China to be a top flight Civ.
Ottoman - I took the Ottomans for a spin (on a continents map, no less) precisely because their UA is so narrow. Not only is it narrow, however, but even when it's good, it's not that good. If you can steal galleys, you definitionally have better boats available. I managed to get four of them quite early, but (and some of this was the properties of my map, which had no shallow channel between the continents), having a navy of that size was mostly just draining my resources, and the galleys were just worse than the triremes I built to capture them. After a while, I stopped trying, since the galleys were worth more sunk as XP than as my own ships. The ability is a lot of fun, but "large early navy" is of only moderate utility. I think that if I had been much more aggressive or on an archipelago map, I would have appreciated them more. Janissary is an above-average unique unit. Sipahi isn't, although I only built one, and mostly for its own sake.
Russia - Opinions seem mixed on Russia, and I can see why. I actually got fairly lucky with having strategic resources in my territory, so I was able to reap the production bonuses, which were more significant early on, especially coupled with the game's attempts to start you near tundra, which indirectly dumps you near more plains. The doubled available Horses/Iron/Uranium was never relevant even though I went out of my way to try to take advantage. I can see iron being relevant if you only get a little, but I had several and was prioritizing extra anyway because of the production bonus the tile gives. Krepost, like most "do something good to a building you want anyway" buildings, is a good - maybe great, although I ended up buying a lot of tiles that game anyway because I had a bunch of gold and was competing with the AI for them. Cossack is not a good UU.
Siam - Yeah, Maritime city states are probably too good right now because the AI won't wipe out ones you're benefiting from and the bonus is crazy disproportionate to the cost as the game progresses. Yeah, making that bonus much larger is stupid powerful, and tacking on a huge culture boost isn't slouchy at all. Siam is nuts. Even if you're only allied with one cultural or maritime city state, by mid game (and even earlier) the bonus blows that most civs get out of the water. The Wat, at +3 Culture, +1 Gold over the building it replaces (a good building to begin with), is a nice bonus on top of the best UA in the game for anyone who's not going all-in on conflict. (And maybe even for people who are -- growing your cities crazy fast and picking up social policies is good for any strategy.)