You had removed the name attribute from your add to cart button and that was used to trigger the add to cart. It had nothing to do with running the checkout wizard.
You could give the add to cart button an ID and set the style to display:none initially and then display it only after the first click in the javascript.
Delete could be accomplished by adding a div tag around the code that gets added with the click and then pass that id to a delete function that can remove it and remove the reference from that added clickArray array.