The functionality that you are looking for is not supported by the DropDownList widget so extending it is the most appropriate way. Assistance related to custom widgets is out of the scope of the standard support service that comes with the controls but you may check this tutorial
as it explains the basics with code examples.
The styling can be easily implemented if you use same HTML as in the original DropDownList. In case you plan to work with the build-in Kendo UI icons check this demo page:
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