How can I change it to present the options to select an options from menu as in the next image below, i.e. I would like the menu to popup where I can select to create a new Word, Excel, etc. document or create a folder?
You need to integrate sharepoint with Office Web Apps in order to see these options on 'New'. This article will guide you.
The "Create New Files" menu appears in a document library when the content types are set to "No" or off. This is found in the library settings > advanced settings > content types.
Creating new document libraries is a good way to test this as the default setting is "no" for content types. Be sure to check what content types exist in your document library before you turn it off and blow them away. Also content types and their templates will show up in the Files ribbon tab under the new document menu.