Assuming you are a US citizen, a Schengen visa will be unneccesary. Holders of a valid US passport are permitted stays of up to ninety days per six month period in the Schengen Region (Austria, Belgium, Denmark, Finland, France, Germany, Greece, Iceland, Italy, Luxembourg, Netherlands, Norway, Portugal, Spain, Sweden) as well as Switzerland (UK is six months annually). Extended stays require visas from countries of interest. Here are the Foreign Entry Requirements from the US State Dept.:
http://travel.state.gov/travel/tips/...ures_1229.html
Most US health insurance plans are invalid internationally, there are, however, numerous firms writing short term travelers' policies offering medical and other coverages (trip cancellation, lost luggage and other travelers' emergencies) as even a rudimentary search will demonstrate.