Risk is positive sum as long as you all collectively agree that since Carl controls all of Asia, most of Europe, and most of North America, he's no longer welcome in the sum.
Yep! I find that many of them can be played with children by just ignoring the scoring; the process of building and expanding is fun enough. At some point, they will naturally start to notice that their line is longer than their brother's, and then you can introduce scoring.
Not quite positive sum, but there's a German board game called Max.
Players cooperate to get a bird, a chipmunk and a mouse safely to a tree while a cat named Max is chasing them. Players roll dice that decide how much the cat or other animals move, and they decide what order to move each, or whether to distract the cat with milk or catnip.
The original Monopoly, not sure how easy it is to get your hands on a modern version but it has two rulesets and you switch from classic Monopoly rules to Prosperity rules when a majority of players agree. The Prosperity rules are non-zero-sum.
I'm not sure I understand the question -- in a sense pretty much all games that have a single winner are "zero sum" in that only one person can be the winner. But maybe I'm misunderstanding -- could you provide some context?
Zero sum means that when one player gains an advantage of +1, another player loses the advantage. Chess is a perfect example.
Monopoly does not count as positive sum IMO because when other players get rich, you end up paying more rent. Cooperative games where multiple players are good vs a few bad ones may be positive sum.
This is perhaps the wrong lesson.