OK. Well featured products first. It seems to be working as it should.
If a category only has one product then the product listing is not shown on your store. So, the store goes straight to the product_info page of that single product. On all the pages that show a product listing (ie more than one product) the featured products are shown.
You can change this behaviour from admin so that the product listing is still shown for single product categories but it is not recommended.
Whether the center boxes appear on product_listing pages and the home page are set in admin>>configuration>>index listing. One setting for the home page. One setting for product listing. If that is not working the way it should then post again but people would need to know what your settings are.
The default settings are
Show New Products on Main Page 1
Show Featured Products on Main Page 2
Show Special Products on Main Page 3
Show Upcoming Products on Main Page 4
Show New Products on Main Page - Category with SubCategories 1
Show Featured Products on Main Page - Category with SubCategories 2
Show Special Products on Main Page - Category with SubCategories 3
Show Upcoming Products on Main Page - Category with SubCategories 4
Show New Products on Main Page - Errors and Missing Products Page 1
Show Featured Products on Main Page - Errors and Missing Products Page 2
Show Special Products on Main Page - Errors and Missing Products Page 3
Show Upcoming Products on Main Page - Errors and Missing Products Page 4
Show New Products - below Product Listing 1
Show Featured Products - below Product Listing 2
Show Special Products - below Product Listing 3
Show Upcoming Products - below Product Listing 4
New Products Columns per Row 3
Featured Products Columns per Row 3
Special Products Columns per Row 3
Filter Product Listing for Current Top Level Category When Enabled 1
If you have things turned off then there should be some zeros in there.