As the options imply in the category settings, there are the latest items from the current author - the ordering can't be changed.

If you wish to show something else, e.g. the featured items from the entire site, you should use the K2 Content module.

An exact URL would help.

You are the first to ask any sort of filtering for what has a distinct functionality.

The fact that this is stream of items doesn't mean it can be ANY stream of content.

The "K2 Content" module is already very powerful and it's exactly for this sort of content retrieval and filtering. There are tons more on getk2.org/extend (free or commercial) by third party developers.

Here's my advice for you:
- You can create one or more module instances of "K2 Content" to distinct module positions and then call each such positions wherever you want using the free Modules Anywhere plugin (from www.regularlabs.com/extensions/modulesanywhere).
- Follow this guide getk2.org/documentation/tutorials/templating-with-k2-and-the-concepts-of-sub-templates to copy the default K2 template overrides set inside your template's structure (if it doesn't support K2 overrides already).
- Then edit the file item.php which is used for viewing the item and simply place the Modules Anywhere plugin tag which in turn calls the K2 Content module instance you've created on the first step.

1. Great.

2. There isn't any because this is inside the component body. That is why I mentioned you need to place the plugin tags of Modules Anywhere inside the actual output of the component body (in item.php). For example in this line: github.com/getk2/k2/blob/master/components/com_k2/templates/default/item.php#L421

You'll publish the module in an unused or new module position (you can add new module positions where you select them on the right side of the module settings) and then you'll add this snippet on line 421 of the file I mentioned:

{module Name of My Module}

3. See 2.

It's not a matter of editing code, it's a matter of inserting a plugin snippet to surgically position something. The same would apply on WordPress for example.

If your templates supports module positions below the component body, then locate that position name and publish your K2 Content module there (remember to also assign the module it on all pages in its parameters).

You're talking to the developer of K2 :)

Noted to consider for future releases.

I said "noted". In the meantime you should use one of the proposed solutions.