EDIT: You don't need a car to allow cross-country mobility; trains are more than adequate. Here in Europe you can travel from Portugal to Finland on rail, and we can visit up to 30 countries for less than $500 with an InterRail pass.
This is the problem. Would cars be worth their expense and inconvenience if there were decent public transportation, walkable cities, or people could generally live near where they worked?
Most of the world doesn't have decent public transport because most of the world is -- rapidly -- developing. In the same vein, most of the world doesn't have decent internet access (I'm not saying the two are very similar in other regards). It's a bit early to say whether the developing world will end up with decent public transport. They're certainly trying with some success. They're also acquiring cars at an astounding rate, they're going to have to do major infrastructure works at enormous cost either way, hopefully they're smart about it.
As for the rest; living without a car doesn't require anybody to live in an arcology or in a "hyperdense" urban area. Most of the world (ie >50%) already lives in an urban area. There are lots of medium-sized or even big cities that aren't hyperdense by any sensible definition of that word. I don't think most people consider it hell on earth.
Having lived in Seattle, Berlin, New York, Montreal, and San Francisco, I would say that Seattle's transit is unusably bad (as is SF's, the other 3 are rather good).
I think the issue here isn't that cars aren't necessary in the States (obviously they are), rather that that's a result of insanely pro-car anti-anything-else public policy in the States, and not because driving is 'naturally' so empowering. It's hard to think of a wealthy country where a city as large as Seattle would have such terrible non-car transportation, yet Seattle is even above average by American standards.
Not without going through Russia. There is no longer any passenger train connection between Finland and Sweden, so that would require either going through Saint Petersburg or crossing the Baltic by ship. You can however do the trip by car; it's just going to cost _a lot_.
I'm now living in Berlin instead of Helsinki, and I take trains to go pretty much everywhere in Europe. When you factor in the time of getting from/to airport, security checks etc. it is often nearly as fast and much more comfortable. As I write this, I'm having lunch near the railway station before leaving for a conference in France. Usually I'd eat in the train but now I happened to have the time to spare...