Display Categories added to Pages?

I’ve successfully added the options for selecting “Categories” as well as a “Feature Image” to pages in my admin panel.
My goal is to do the following:

I am setting up a category for a “portfolio” with sub categories for different kinds of work (ie. web design, graphic design, photo retouching etc.)

1)I would like to be able to display all the related sub-categories & parent category (Portfolio) on a page as thumbnails, however they are not showing up when I view the category as a page, even though the pages are registering as posts under the category names.

2)I would also like to add three randomly selected sub-categories as thumbnails on a given portfolio page under a heading of “Related Projects”.

Is this possible? Any direction would be much appreciated.