Adding a menu or dropdown to Quicktags API

With Quicktags, no. It’s for buttons only, and only in the text view, not visual view. But it is possible to add all manner of UI elements to the TinyMCE editor, though it may in some cases take writing a TinyMCE plugin to do so. In many cases, UI elements are already available, but are hidden so as not to clutter up the editor area. There’s all sorts of articles on customizing TinyMCE, here’s one to whet your appetite.