The golden rule is to segregate. Give cars, bicycles, and feet their own space. DO NOT mix. If somebody can make a choice to put their 6 year old child on a bike and cycle around a city, you've got it right.
My city of birth and upbringing (Warsaw, Poland) has a huge problem of people not respecting each others' space. Unfortunately a lot of bike lines are just painted on sidewalks. This (and the fact there's not enough bike lanes) encourages cyclist to just drive around pedestrians, who sometimes also tend to drift onto dedicated bike lanes.
It's terribly annoying to travel on foot during summer/autumn. You can easily get hit or at least shouted/honked at by a 'deranged' biker. Pedestrian/car and Cyclist/car relations are even more complicated.
Now living in Berlin I have to say this city has mostly solved the issue. Majority of bike lanes are a (separated) part of the car lane and cyclist are encouraged to just mix with cars on low-speed streets (<30 zones). Annoyingly enough some (not many though, mostly deliveries etc.) cyclists completely disregard the signage and still ride on the sidewalks.
Still, the roads seem to be really safe, hence most cyclists don't even wear helmets. I noticed drivers in here also pay more attention to what's happening around them. You rarely read about serious accidents involving bikes. Last one I remember it was a case of road rage where car driver beat a cyclist with a bat.
But also, Seville has one of those easily under-estimated features that engenders a great cycling experience, and consequently good chance of success with such a plan - it's relatively flat. Amsterdam enjoys a similar benefit, though I can't speak for Copenhagen on this aspect (those were the two extant 'cycling capitals' mentioned in the article).
Thinking about it, there is some truth to this. I have kids and would NEVER dream of cycling with them in Paris; we put the bikes in the trunk of the car and go to a big park if we want to ride.
If there were actual segregated bike lanes where there is zero chance of crossing a car (at any speed: I mean NO car) I might cycle with them in the city, and I admit it would be nice.
Also "mix everything" is a viable option, too, if people behave themselves. The "woonerf" (http://en.m.wikipedia.org/wiki/Woonerf) is a very good solution for densely built neighborhoods (even in the USA, it seems: http://ithacavoice.com/2015/04/ithaca-housing-project-includ....
In a similar fashion, I see little wrong with cyclists driving on sidewalks. Problems only start when they drive faster than pedestrians in a busy street. In other words: if they do not acknowledge that they are guests on the sidewalk.
That is fairly typical behaviour in the Netherlands. People will cycle through almost empty pedestrian-only zones at 10-15 km/hour, slow down to walking speed when coming close to pedestrians, and start walking, bike in hand, if things get really busy. Legal? Not everywhere, but this rarely gets you a fine.
This seems too emphatic, given the counter-example of Japanese cities, which have few dedicated bicycle lanes, yet remarkably high rates of cycle usage.
The Japanese shifted liability onto larger vehicles. Just as most places will always treat a rear-shunt as being the fault of the car behind, hitting a cyclist is always the fault of the larger/motorised vehicle.
Having segregated (and physically separated) paths for bikes works well though. They remove the fear that when you're riding straight, a lorry comes from behind and kills you outright. Here the paths are usually shared with pedestrians, and while pedestrians are annoying, they are few (compared to Japan...) and it works reasonably. Intersections are a bit of a problem, but the mistakes by car drivers are quite predictable (and being also a driver myself, I know why they happen), so I can watch out.
I guess one needs to match effectiveness of incentives with the local culture.