I'm saying as a game mechanic it's one more thing where it's really easy to get negative modifiers and really hard to get positive modifiers.
I rarely have more than one or two civs covet my lands in any given game, and partly it's dependent on civ personality. On the other hand, in G&K most civs other than your immediate neighbours will automatically give you the positive "You have no contested borders". Also, the "covet lands you own" is no longer an immediate alert that war is coming - civs will drop it if they like you, and at least in the game's early stages it's almost always a weak (dark) negative rather than a strong (bright red) one.
In one recent game Ramesses coveted my lands, sent an army my way and became hostile. But after I made friends with Babylon and India, and joined them in denouncing Arabia, he decided he didn't want to be left out in the cold, and offered a DoF. I didn't fully trust him for the rest of the game, but he stayed true to the friendship - the game ended with a joint war against Polynesia.
I like to interact with the other Civ's. In this game they make it pretty hard.
I don't find that at all. I almost always have at least one neighbouring civ rapidly offer a DoF; sometimes that snowballs into games where I'm at peace with everyone game-long (except for wars I start, or agree to join in). Certain civs - like Babylon - I can rely on to be gamelong friends if I'm on their side from early on.
It's also very feasible to target a civ for favours and change their attitude towards you. In one game I was given intrigue that Harun was planning to attack me. I promptly denounced one of his enemies. We entered into a DoF sometime afterwards that lasted the rest of the game.
Bear in mind that the most important diplomatic features in the game are declarations of friendship and denunciations - as I note below, the AI places more stock in denouncing the same enemies than it does in going to war against the same enemies. "Denounced the same leaders" is a strong, long-lasting penalty that all civs will almost always view very favourably unless they really hate you. "Declaration of friendship with the same leaders" will almost always lead to a DoF offer from the target civ and the removal or mitigation of negative modifiers. It's a little less pronounced in G&K than vanilla, but it's still generally the case that civs care more about your behaviour towards the civs they like or hate than they care about your behaviour towards themselves.
If you have a way of sharing China's religion, or spreading yours to the majority of their cities (if they don't have a religion themselves - if they do they're likely to get annoyed with you competing with them, although I've only ever had that modifier in one game), that's also a strong positive for most of the game, and remains important even in the later eras since it's a permanent modifier, if a slightly weaker one late in the game.
But if you want to save China, all you need to do is actively fight units attacking Chinese cities. I did that with Neutral Mongolia in one game (who I think may have coveted my lands, and possibly also my Wonders), more to weaken the Hun attackers than to save Genghis Khan. It took a long time, but Genghis did eventually offer a DoF, and that too lasted the whole game.
I could save China if she'd let me, but she won't. And I've learned from experience that 'attacking them to save them' doesn't work.
There's an issue with a "missing" modifier - you can't denounce a civ you're at war with, however if a civ you like is at war and you denounce the mutual enemy, you don't get a positive. This makes no sense when you get positives both for denouncing someone the civ you like has denounced, and for going to war against a common foe (the former stronger than the latter).
Common foe is a strong positive, but usually not a particularly long-lasting one and there seem to be substantial differences between civs in what you have to do to earn it. Wu is not easy to impress; with other civs it can be fighting against the same enemy units, with others just joint declaring war with them is enough.
Everybody just hates you and you can't say, 'But I had to attack China to get to the real bad guys!'.
I'm not sure why you would do that. Most civs will offer open borders (if you have a giant army nearby you can even get a positive when the civ you want to save asks you if you're planning to attack them and you say no, as long as you keep that promise); if they don't like you much they'll demand you pay them or give them luxuries, but it's very rare for a civ that isn't openly hostile to refuse any open borders deal at all. Warmonger penalties are erratic - sometimes everyone hates you, sometimes they don't. In the game I just started I accepted a Maya offer for a joint war against Babylon. Of all seven other civs in the game, only one civ gave me the "warmongering menace to the world" modifier (and it wasn't Babylon, even after I made peace - in fact they promptly offered a declaration of friendship and, being Swedish, I wasn't going to say no. Sure, giving a potential rival a 60% boost to Great Scientist production could be risky, but the AI generates GSes much more slowly than a human anyway, and I know from experience that Nebuchadnezzar is a reliable ally - giving him a tech boost helps me), and that was weak.
If I take a Chinese city that was taken by Rome is there a 'liberate' option that lets me return the city to the Chinese?
Only if China is extinct. If they aren't, you can give it back to China in the trade window. That only gives a "You traded recently" positive, but it will be a strong one.
Should be moved to Rants thread... If you were playing multiplayer and someone wonderfailed you on a wonder you really wanted for your strategy, would you love him for it, would you be indifferent to him, or would you hate his guts and possibly want to take the city with the Wonder if you could?
I've rarely seen this modifier in G&K, but when it was ubiquitous in vanilla it annoyed me - mostly because it's bad design. An AI is not a human and doesn't have human considerations. An AI playing to win does not play to win the way a human does, and this difference has to be taken into account in programming modifiers. This is particularly the case since the AI doesn't really understand Wonders very well.
Take your analogy: the other player builds a Wonder you really wanted for your strategy. Would you want to take the city with the Wonder if you could?
First question: what's the Wonder? Would you want to take the city with the Great Library? The Oracle? The Hagia Sophia? In almost all cases, no.
Would you want to take the city with Hanging Gardens, Chichen Itza or Notre Dame? Quite possibly.
Would you want to take the city with Petra? Very much depends on whether it's a good "Petra city" or whether the other player just built it in flat desert.
The AI can't discriminate - if it's programmed to want the Great Library, it will hate the player who gets the Great Library, a modifier that increases its chance of going to war to capture a pointless Wonder. It also doesn't have enough of an overarching concept of strategy to follow through - so it might be prompted to go to war by a negative "You built Wonders we coveted", only to attack cities that don't contain that Wonder anyway, because the algorithms that determine which cities it wants to attack aren't related to those that define its target Wonders. And yes, this can lead to situations like the OP's, where an AI ends up taking actions detrimental to its chances because of a badly-designed modifier, whether it's wasting time (and accumulating warmonger penalties) by launching wars it doesn't gain anything from fighting, or refusing open borders to a civ in a position to go to war with its rival.
a bit more constructive please. 
O.K., seriously, if you can get to Rome without the OB, you could fight them and rack up some significant positive modifiers. Soon she would warm up to you. I don't know how long it would last but I can almost guarantee it would happen. If she turns on you then it's back to the "war game" approach. It's an acquired taste.
