The menu item should be set for logged in users only (obviously). Additionally, you need to create a K2 User Group with the right frontend editing permissions and then assign your current Joomla "Registered" users (or whichever other group you use for that matter) to that K2 Group, though the K2 Users list (select the users and batch-assign them to the relevant Joomla and K2 user groups).

As for your second question, that's default behaviour. K2's item form has to exist in its own context, it cannot be inline with the rest of your site content to avoid CSS/JS conflicts.

A good example to see how such frontend editing access can be "orchestrated", signup at getk2.org.