Follow these steps:

- Enable Frontend Editing in K2's component settings.
- If there is no K2 User Group already in place for "Registered" users, create one and adjust editing permissions, as well as which categories you'd like them to add content in. By adjusting these correctly, you can control how users submit content, e.g. do they self-publish or await moderation by an admin?
- Go back to K2 Settings and adjust new user signups to be assigned to that K2 User Group.
- Publish the K2 User module to allow users to login and see their frontend editing options.
- Optionally create menu items that point directly to the item edit form, for quick content submission. Make that menu item available to logged in users only.

That's pretty much it.

You can always signup at getk2.org which uses K2 entirely for content submissions for extensions, tutorials etc. to see how it's all setup.

You can use a module template override to change that menu and use different phrases as well.

See this getk2.org/documentation/tutorials/templating-with-k2-and-the-concepts-of-sub-templates for overriding K2's templates.

Do a module override as I told you. It's pretty simple.

The Joomla user login module is irrelevant to use.

I'm afraid that's not possible. Explore perhaps the option of using some form extension which allows content to be submitted into K2 by anonymous users (e.g. RSForms or something similar).