As stated, the servlets need to be instantiated using the nullary constructor, and so you cannot define a non-nullary constructor and expect it to be used. Any initialization that relies upon any context information must wait until the init() life-cycle method is called. You could, I suppose, perform simple initialization in a nullary constructor, but to do so is counter to convention and therefore a poor practice.
The init() method is typically used to perform servlet initialization--creating or loading objects that are used by the servlet in the handling of its requests. Why not use a constructor instead? Well, in JDK 1.0 (for which servlets were originally written), constructors for dynamically loaded Java classes (such as servlets) couldn't accept arguments. So, in order to provide a new servlet any information about itself and its environment, a server had to call a servlet's init() method and pass along an object that implements the ServletConfig interface. Also, Java doesn't allow interfaces to declare constructors. This means that the javax.servlet.Servlet interface cannot declare a constructor that accepts a ServletConfig parameter. It has to declare another method, like init(). It's still possible, of course, for you to define constructors for your servlets, but in the constructor you don't have access to the ServletConfig object or the ability to throw a ServletException.
Bottom line. Servlets follow the JavaBean architectural paradigm. One of the core principles of JavaBeans is that they should provide a no-argument constructor method so that a simple generic brain-dead BeanClass.newInstance() can be used to construct the bean without regard to environment or context. Environmental/contextual considerations should be managed post-construction. In the case of servlets, that's the init() method.
An IDE is no substitute for an Intelligent Developer.