Great questions, thank you!
Yes, communal living is pretty normal here, even in the cities. Communes are not unsual here, and you find people from all walks of life living together in communes. I believe commune living is so sough after that there's actually a lack of communes, considering the demand for them. In general, Danes are community-oriented, and you find that most municipalities always arrange local events and activities for families with children and the elderly, in particularly. It's super normal here to be part of various social clubs and groups, particularly sports clubs, where people can have fun, socialise with like-minded and establish friendships and networks.
It's interesting to me that you noticed Danish children's independence. I think it's a core value in most Danish families to teach children responsibility and independence from an early age. Again, when I read on Mumsnet that parents do school runs, even for older children...that's so foreign and incomprehensible to me. Most kids here get to and from school by themselves. They go to school as locally as possible, and they learn the traffic rules at a young age so they know how to navigate the streets and lanes on their bikes. Of course some parents do drop their kids off and pick them up, but I wouldn't say that's the norm. You also often see kids go to and from school with other kids / their class mates.
As for the dish you describe, that's rød grød med fløde, haha. Danes like to ask foreigners to try to pronounce it because, as you mention, it's impossible for them to do it - and Danes think it's absolutely hilarious to hear foreigners struggle to say it, lol. But in reality, no one really eats rød grød med fløde. Maybe the older generation and babies, but probably not even them! I actually do like it, though, It's a strawberry purée with cream poured on top.