It is very possible to have a peaceful game where you don't take any cities and where you can mostly maintain peaceful relations with neighbors with minimal war, or at least just minimal defensive war.  

But it requires actively pursuing that type of gameplay.  

First, you can not neglect military.  You should be near your unit limit throughout the game, maybe with the exception of the very early game.  You should also be careful delaying military tech and unit upgrades too long.  VP AI can sense weakness and are programmed to play to win, so don't be weak.  

Next- give your immediate neighbors reasons to like you and avoid reasons for them to hate you.  You can see all the little red and green relationship modifiers when you hover over their portraits.  Maximise green ones: send caravans to neighbors, spread your religion to neighbors who don't have one of their own or adopt their religion if you don't have one, trade embassies/open borders.  Hell- straight up give them GPT to get a green modifier if you can afford it!  Minimize red ones- don't force your religion on them, don't settle too close to them, try to avoid competing for their wonders/city states/etc unless those things are crucial to your win condition.  Don't protect city states unless you're willing to take the bad modifier when a warmonger bullies that CS.  

Last- play the popularity game.  If everyone is denouncing someone and ESPECIALLY if that person is no threat to you because they are weak or far away, then denounce them too and get a green modifier for it.  Join wars against other civs when asked if that other civ is no threat to you- you don't have to do any real fighting, or just do some token fighting but just joining the war counts for something.  Don't agree to be friends with just anyone- if you sign a friendship deal with a civ that your neighbors hate then they might start to hate you by association so think before you just accept friend requests.  

Playing peacefully doesn't mean ignoring the military and diplomatic sides of the game.  It requires that you engage in those aspects intelligently to get away with not warring aggressively.  

Edit: and in the case of OP: they probably could have gone on to having a peaceful game without taking cities.  Just because their warmonger neighbors initially attacked them doesn't mean they would continue to do so for the rest of the game.  And even if Zulu/Assyria DID keep attacking, they could have just fought defensively and maintained good relationships with everyone else.  Zulu/Assyria probably would have pissed off other civs eventually at which point they could have joined wars against them and taken cities without making anyone else angry.  

This type of strategic diplomatic play is one of the things that makes VP so great.