To get an un-styled cart, simply remove the cart .css file. Completely un-styled would also remove the mobile friendly media queries. It might make more sense to delete all but the media query portion of the .css file to get it un-styled.
All of the actual styling is done in the .css file. The class="" in the html is also used as identifiers.
I can't debug an attached page like that. I'll need a live url to view the page, since the php, html ,and javascript all work together to make a functional page. If you give me a url I can test it that way and tell you why it isn't working.