Yes, similar policies are practical for larger nations to adopt. In the United States for instance, Boston's population is about the same as Copenhagen's. Given the political will and the availability of funds, there should be no practical obstacles to the introduction of special lanes to encourage the use of bicycles.
Yes, cities need to make changes in order to improve the air quality, address climate changes, reduce traffic, and improve its citizens' health. Riding bicycles is a great way to address all of these problems, as Copenhagen has successfully shown through its own bicycle program. This is a low-cost, high-impact strategy.
There are lots of ways for countries to make it practical for more people to ride bikes and less people to drive cars. While this is more practical in urban and suburban areas than rural, there are many ways to encourage people to ride bikes instead of driving and many American cities are having success with this.
Copenhagen is a relatively small city, area wise. While use of bicycles there certainly seems to work there are significant hurdles to overcome in order to implement such a measure in a larger nation such as America. American cities are much more disperse and widespread. This means greater travel distances and larger goods being moved. Things that simply cannot be done on a bicycle.