K2 does NOT do that for sure. You need to provide more info on your setup (K2 & Joomla versions) as well as any system plugins that *could* affect your menu system.

This is intentional. But this only occurs for the item edit form, because it cannot (and should not) be integrated into a site design for practical reasons.

Use what K2 offers now. Publish the "K2 User" module in some position and all user actions for adding/editing content will be available to your users. Just make sure you setup the right permissions in K2's User Groups (in the backend component).

getk2.org is a great example of content being published by users. Login and have a look. You'll see the editing process is pretty simple & smooth :)

K2 *has* to open the item edit form in a modal because a) that form cannot fit a template's design and b) we don't want the template's CSS/JS to cause issues to the edit form.

You're not doing anything wrong. It's how frontend editing works in K2.

Whatever suits your needs.