I was looking at this example of an admin theme (http://themepixels.com/main/themes/demo/webpage/shamcey/dashboard.html). As for the Left Admin bar labeled "Navigation," Is there a way to achieve this type of Left dashboard Admin menu without using a plugin? I would like to achieve this with css, js, or Jquery anything dealing with coding it out.
You have to put the code somewhere. That is either going to be a plugin, a must-use plugin, or a theme.
A theme is a somewhat strange place to be putting code intended for the backend, so the answer is that you probably need a plugin whether that plugin is one you write or one you get from somewhere else.