The module has its own layout, different from category layouts. By default it presents the items in a list (vertical). The layout settings in the category only control how the items would show when viewing the K2 category alone.

You need to add some CSS code to "break" positioning into 4 columns. I could give you the CSS changes, but you need to have your site online to check it out.

You can create overrides for both the component views and the modules.

Compared to regular Joomla overrides, K2's are a bit supercharged: getk2.org/documentation/tutorials/174-templating-with-k2-and-the-concepts-of-sub-templates

For the K2 Content module, simply copy the "default" folder from /modules/mod_k2_content/tmpl/ into /templates/YOUR_JOOMLA_TEMPLATE/html/mod_k2_content/. The same applies for the other K2 modules as well. You can even create multiple sub-templates for K2 Content specifically.

Finally, if you copy /components/com_k2/css/k2.css into /templates/YOUR_JOOMLA_TEMPLATE/css/ you'll also be able to totally override the default K2 CSS. If you want to start fresh with your own CSS rules, you can disable loading k2.css in K2's component settings.

Overall, K2 is the most flexible component to override in the Joomla community (even compared to core Joomla components).