IntelliJ IDEA does not provide any intelligent support to Spring Initializr projects?
I created a Spring Initializr project using the Spring Initializr built in wizard in IntelliJ IDEA 14.1.1.
If I right click a package name, then go to new, there is no choice for creating a Java class in the context menu (see the screen shot bellow). 
Like the same, there is no way of creating a jsp file in the WEB-INF folder.
What am I missing?
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Did you create is a Maven project? Just click on "Add as Maven Project" in the Event Log:
After importing all the folders will be configured correctly.
Yes, "Java Class" choice is now there in the context menu.
But there is no choice for creating a jsp file in the src/main/webapp/WEB-INF/jsp folder. (this problem is noly for Spring Initializr projects)
You need to create a Web Facet and setup web roots, see https://www.jetbrains.com/idea/help/enabling-web-application-support.html