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The missing city you are probably looking for is San Francisco...it has the bullet points you want more so than the other cities. Not that there is anything wrong with the other cities I like Chicago/NYC/Toronto/Seattle a lot, less for LA.... just don't think it is what you are looking for.
San Francisco and DC (Toronto,too, minus the weather) are more like Melbourne, dense, cultural, diverse cities with a lot to do. They are are also closer in climate to Melbourne.
But if you're Finnish, i don't feel that the cold weather will be as big of a shock to you.