It is possible to have MathJax renderin drag and drop elements. If you have the MathJax filter enabled, you can add filter_mathjaxloader_equation class to an html element containing . You may also have to include some delimiters elsewhere so that the filter loads MathJax.
Drop down menus are specified by HTML standards to be text only. It is possible to configure MathJax to render that anyway, but it will only be visible in certain browser. This makes it a very bad idea to use.
The TeX notation filter is not necessarily a bad solution if MathJax fails for some reason. It is just much more difficult to configure than MathJax which works out of the box.