Wordpress - Pulling in dynamic category images to a li

I'm running into a problem pulling in category images in dynamically to a li. Im kinda new to Wordpress and have no issues building the theme but sometimes run into issues pulling in the content I need. Heres what i'm looking to do.

I have a section on this site(http://ift.tt/19p2XTy) "News Topics". Each icon (image box) will be a category and the image will be associated with the category. Im going to use ACF to attach the image to the category but I need to be able to pull those images into the li dynamically and each time a new category and image is added in the back end, it needs to just automatically load on this list. I also need to be able to EXCLUDE some categories that don't belong on their. For instance the homepage hero slideshow is loaded via a category, so I would need to leave that category off the list. When that image is clicked it also needs to load the category title and description in the blue box below.

So this is my problem i'm trying to solve. If anyone could steer me in the right direction I would be very grateful.

Thanks -Dan