Seems like no matter how strong a friendship I form with civs in the early game, almost all of them fall apart by the time the congress gets going, regardless of how little conquest I've done. I was wondering if this was by design or if there's something I'm doing wrong.
There are quite a few factors. You have to know what makes each civ tick, some of which will be shown in the transparent diplo information. A warmonger with no other neighbour will attempt to kill you, no matter what you do. Diplo civs like Austria, Siam etc. will hate it if you have many cs alliances. Similar with religion.
AI tend to gang up on weaker civs, make sure it's not you. You can trade small favours, but most importantly, you should watch very carefully who is friends with whom, who denounces whom and then choose your side, or stay out of it. Early joined wars or denounciations can work wonders for a good long term relationship, even if you have a certain amount of border dispute or religious rivalry. Often friendships work in triangles, watch out for those. Some civs are just a little bit wary of upsetting several civs at once.
If you're going domination, you might not care too much about anything, but even so, the enemy of your enemy is your friend. So you might befriend them and kill them last as a reward, or swiftly, that's also ok.
When you get ahead or close to winning, civs can turn on you. Usually though, you are then either ahead in tech, especially military tech and can defend yourself. Or, you have many cs alliances or even defensive pacts which will makes civs think twice about attacking you.