Not all Web parts are using the templating infrastructure. As a rule, only the "repeatable" Web parts (blogs posts, comments, wall notes, etc.) are using templates. If you need to change the appearance of such controls, all you have to do is to change the existing templates (in App_Templates/[your theme name]/) or add a new template and use the part settings to point to it. Templates are simple HTML or ASCX files with placeholders that are used by MonoX to inject the dynamic content.
Changing the appearance of simple parts like login is simple. Just create the new Web part and copy the markup from the LoginModule.ascx to it. In the codebehind (ascx.cs) say that your new module inherits from MonoSoftware.MonoX.ModuleGallery.LoginModule. Now you can change the front-end markup as you wish - please don't remove the existing controls from it (as they could be referenced from the codebehind); instead, you can set their visibility from code.