For example, they get mad at you when you ask them to stop spying at them, and also get mad when you catch one of their spies. They get mad when you vote against their propositions and vote against yours all the time. They spread their religion all the time but doing the same to them gets them mad. They settle near you and asking them to stop makes you mad, but god forbid you settle near them. Is it possible to disable this? I'm open to mods.
Unfortunately the AIs in Civ 5 are always looking out for themselves and are unlikely to make any effort to not disrupt you. If it benefits them they're going to do it.
The only things you can really trust they wont do is go to war with you if you have a Declaration of Friendship (unless of course you attack first). You can't even count on them to vote for you in the World Council unless you buy their votes.
I don't have any evidence of this, but I think some of it also has to do with the AIs personality. Agressive AIs are probably more likely to break promises, while diplomatic AIs are more likely to follow through with a promise for its duration(30 turns). After that you're fair game again though.
I'm fairly certian I've even caught spys in my citys from Civs I was allied with...
Unfortunately, this behavior is by design, and there's no real way to mod it out of the game as far as I'm aware. The lead designer for Civ 5 even lamented this in a Kickstarter update for his post-Civ5 game:
The AI plays to win, and all of its choices are designed around what gets it ahead in the current situation. In practice, this means that the AI is going to react aggressively to everything it views as threatening, and will turn right around and make threatening gestures towards you, while not caring if it upsets you in turn.
There's very little give-and-take with the AI. Plan on war frequently. Don't play to appease the AI - there is no appeasing it for any length of time.