Adding drop-down menu to chrome extension icon

You can't.

You can either register a click via chrome.browserAction.onClicked but show no UI, or open an HTML page in a popup. You can style it to look like a menu, but it will still not be like a native drop-down menu.

Note that you can right-click the extension button, but the menu you get there is fixed. I submitted a feature request a long time ago regarding that, but it never took off. There is a contextMenus context "browser_action" now that achieves this.