Correct. K2 is a Joomla Content replacement. And it's much nicer to work with compared to Joomla articles.

So you just work in K2 and create menu items pointing to K2 content.

As for the menu example you provided, you're linking to a single category in K2. When you do that, K2 will fetch the display properties from the K2 category. Unlike Joomla content, K2 follows a more natural approach to category parameters and inheritance.

On the other hard, if you created a menu item pointing to multiple categories, the greyed out fields would be enabled so you can have a uniform layout from items coming from different categories. Imagine a content section that is a 2-column grid and a blog that is a 1-column grid. If you wanted to fetch items from both of these categories, mashed together, you need to redefine the display grid. Everything else comes for each K2 category setting (e.g. whether the K2 image will be shown or not).

Parsing thousands of files could cause the server to reach some limits, in which case PHP would just throw a blank page. Set error reporting to "maximum" in Joomla's Global Configuration and check again. If you see an error, let me know.

Update to K2 v2.10 (dev) from github.com/getk2/k2/archive/master.zip and check if the error still occurs.

Great.