Do check into this Schengen nonsense before wasting too much time and effort. It seems to me this Schengen visa nonsense only applies to non european countries that are required to get a visa to visit almost any european country. For non european countries that do not require a visa to visit european countries, this Schengen nonsense does not apply.
Check on the internet if any of the countries you plan to visit require USA citizens to have a visa. Almost certainly they do not require USA citizens to have a visa. Therefore this Schengen visa nonsense is just that for USA citizens.
If you are a citizen of Tibet for instance, a country that probably is required to get visas to visit any and all european countries, then getting this Schengen visa will allow you to most likely visit all of the european countries with just one visa. You don't need a separate visa for every single european country.
Here is a Denmark website I found that illustrates the above.
http://www.workindenmark.dk/Visa_before And as you can read, if you are from the USA, you don't need a visa of any kind. But for those requiring a visa, the Schengen visa will work and allow you to visit other countries in Europe.
Visa
Nationals from many countries can enter Denmark without a visa
Nationals from Nordic countries and EU/EEA countries can enter Denmark without a visa. Nationals from the following countries are also exempt from visa:
Andorra, Argentina, Australia, Bolivia, Brazil, Brunei Darussalem, Bulgaria, Canada, Chile, China (only people with passports issued by "Hong Kong Special Administrative Region" and passports issued by "Região Administrativa Especial de Macao"), Costa Rica, Croatia, El Salvador, Guatemala, Honduras, Israel, Japan, Malaysia, Mexico, Monaco, New Zealand, Nicaragua, Panama, Paraguay, Romania, San Marino, Singapore, South Korea, Uruguay, U.S.A., The Vatican, Venezuela.
Nationals from all other countries need a visa.