WebDriver deals with HTML, CSS and Javascript. There is no such thing as Date Picker in HTML, CSS, Javascript. There are plenty of libraries which will use the three technologies to create a date picker but how they implement this determines how WebDriver is going to be able to interact with it.
Bottom line, how to use a date picker is COMPLETELY dependent on how the date picker was implemented. If you know how to write a date picker you can usually look at the current implementation and figure out how to use it with WebDriver. If you are a hacker (in the good sense; i.e. learn by trial and error) you can usually figure out how to use the date picker with WebDriver. Final option is to have the developer who selected the date picker library help you with having WebDriver interact with it.
For example, if you look at source for
http://jqueryui.com/datepicker/#inline you will see the date picker is actually a set of DIV and TABLE elements. I would write a WebDriver library which handles interacting with this date picker.
Additionally, WebDriver deals with the HTML code. A screenshot of the application really gives little information. Whenever I am implementing WebDriver code I look at the application HTML/DOM. Without tools like DevToolbar (F12 on IE) or Inspect in Chrome I would not be able to use WebDriver effectively.