People do get long stay visas for 6 month periods, or longer, but as other alluded to, it is a process you engage in with a specific country, they will want to see financial viability, health insurance, and the intent is that you are resident in that country for the visa period (shown by a lease or other means). Travel to other Schengen countries is generally allowed, depending on the conditions of the visa. Most find the effort, expense, and restrictions to not be worth it for tourist travel. If you are serious, pick a country and contact their embassy or consulate here in the US.
Yes, there are Bi-lateral agreements recognized as part of the Schengen agreement, Denmark was mentioned, the US also has agreements with 9 other Schengen countries, including Portugal, Spain, France, Belgium, the Netherlands. About all I will say is that despite these agreements, the fact that you rarely, if ever, hear someone taking advantage of them, speaks volumes.
That leaves what most do, plan sufficient time in non-Schengen countries. This would include the UK and Ireland, Romania and Bulgaria of the EU countries, Serbia, Bosnia, Montenegro and other Balkan countries (Croatia no longer though), then nearby countries like Turkey, the Middle East, and North Africa.