I just won a cultural victory on a standard size map with 16 civs, on normal speed. edit: King difficulty - pre-patch I played Emperor but bumped it down post-patch, will probably go back up again soon though.
I was never at war, never denounced anybody, and was never denounced by anybody. Every single other civ was friendly with me - Songhai were hostile when we first met right at the start, but turned friendly and kept bugging me for pacts of friendship. Mongolia were "guarded" in the renaissance but were then wiped out, and for some reason Siam turned "guarded" right at the very end, but other than that all 16 civs were "friendly" the entire game. I only signed one friendship, with Russia early on, but refused all else. Very rarely signed open borders, and usually only if it both directly benefited me and was with a distant civ. Signed heaps of research agreements, more for the science than the diplomacy, until about the renaissance when I became 100% focused on culture, and had 2 spare sugars which I traded liberally. If Russia asked for anything to honour our friendship I gave it to her. edit: also I only had 2 cities the entire game, but I always play with very few cities, on very crowded maps. Pre-patch I would play with 20 civs on a standard map, but I dialled it back while I get used to the new diplomacy. I was worried that my capital might be a juicy target for greedy AIs as it had heaps of wonders in it, but that never seemed to happen. I was comfortably top of the scoreboard until about the renaissance when I started to slip and by the end I was in about the middle.
You are definitely supposed to be selective about friendships. That's why they ask you for stuff. If you don't want AI always pestering to give you stuff, don't sign friendships. Plus if you ever end up going to war with a friend, it's pretty much game over for you diplomatically. I think that is one of the main reason that relations always collapse by the end of the game - one AI betrays a friend, which causes other AIs to denounce it, including its friends, which then causes other AIs to denounce them and so on and so on. I would like to see AIs much more hesitant to denounce friends (and by the same token, more hesitant to sign friendships).
I still prefer the pre-patch diplomacy (of which I was a vocal supporter) because the post-patch diplomacy is so linear, you can't cancel anything so you can't have shifting alliances, only a gradual entropy towards an all-out brawl. The only thing you can really do is either pick a side and pray, or do what I did and try to stay out of it completely. Making co-operation public is good and having a denounce option is fine, but I miss pact of secrecy. It felt more like you were actively manipulating relationships, where if you wanted to go to war with a civ, you'd ask around for people to sign pacts of secrecy so that they'd be on your side. If nobody would sign a PoS maybe you would re-think going to war. Post-patch all you can do is denounce them and hope that your friends denounce them too. Then from that point on you are stuck denouncing them for the rest of the game.
If PoS was added back in, DoF were cancelable, and Denouncing was made temporary, then I would love the diplomacy.
edit: one other thing - I never built a single trading post either! So I seem to be doing everything wrong according to the general view of Civ5. I am a dove-builder and thoroughly enjoy my peaceful victories.