Wow you seem to be getting really good diplomacy results (this is the most positive view on civ5 diplomacy I've seen) - what map type, size, level do you play, play style/victory conditions? I'm trying to get a better handle on what makes diplomacy tick.
I'm finding the diplomacy ok most of the time but quite flaky. I've had one game (small pangea) where its all gone wrong with friend backstab followed by chain denouncements like alot of people get. I also had one game (tiny continents) that was ended in a lets all be friends lovefest.
Is the trick to make sure your friends are always busy fighting someone else?
You have to be conscious about what you're doing. If you declare friendship with another civ you might as well consider it an alliance.
That means you both should be working for the greater good together, whatever that may be and you should be careful that you don't work with the wrong civs. Thats it. Choose your friends wisely as you must be willing to adopt their enemies AND their friends. What is your relationship with their current allies/friends? Expect your enemies to cause problems for you, and if your friends with a friend of your enemy then its only natural you get backstabbed.
If another civ gives you trouble you should try to get your friend to hate them as well so that you don't run into this problem of them liking people that you don't. They will resist backstabbing you if you get them to hate your enemy before your enemy gets them to hate you. Good news is its easier for you since you're already friends with them, but you still have to do it or they will weasel their way into your alliance and turn them against you.
Diplomacy in this game is interesting if you're willing to see it in its simplicity. I've even gone as far as taunting an enemy into declaring war on me so that it would look poorly on him (and be easier for me to turn everyone on him).
Also, yes if you start winning by a big margin even your friends may turn on you. But what kind of friend are you to be leaving your friends in the dust like that? Think about it... you simply gave them only one option (you can't be surprised when it happens) - if you want to win you have to kill me and what good is the AI if it isn't trying to win?
I think the big problem with Diplomacy is people underestimate how many things the AI is looking at when its making decisions. Yes its flavor is on a scale from 1 to 10 but its more involved in the information gathering than that and the AI will, in fact, hold grudges and try to bring you down over the long run AND try to turn your friends (allies) against you.
The other big problem is the AI over estimates its own tactical ability which undermines its influence on the diplomacy table. Its hard to take an AI seriously if all it can do is make empty threats and lend worthless assistance.