Some more points:
and added a new servlet with name NewServlet to the project
http://localhost:8080/p3/NewServlet looks like it should run this servlet. The URL for having the welcome file served would be
http://localhost:8080/p3/ (assuming that "p3" is somehow the name of the web app, which I have no idea about how that works in Eclipse).
and copied the servlet-api.jar to the lib in the WEB-INF folder
You should not do that. The Servlet API jar files are part of the servlet container; they must not be part of the web app as well.