In this section:

Why don't products appear when I select a category from the menu?

You need to select these menu items to point to a category. Go to the My Store > Navigation > Main Menu section and tap Add More to add a link to a specific category.  

How do I change the color of the cart button? 

The color of the click button is part of the theme design that you have chosen for the entire website and the colors are preset. It is possible to change it from the CSS / JS editor of your theme, which is located in the My Store > Theme section.  

To add a style you need to enter code enclosed in a tag:  
     Your CSS code