leaving comments only for logged in users

A lot of this is built in…

settings -> discussion = Users must be registered and logged in to comment

this setting checked would mean someone has to fully register to your site before they can comment

WP has a meta widget built in you can add to your sidebar which adds registartion and login links, from appearance -> widgets

as for banning
https://www.ads-software.com/extend/plugins/ban-hammer/
Is one option, there are a variety of plugins depending on your exact needs

Thank you Voodoo.
settings -> discussion = Users must be registered and logged in – works strange. It only has the name and password fields for old users, but no option for new users registration.
Besides, it lets the user get into my admin panel, and i dont want it!

To stop users from seeing the admin panel: https://www.ads-software.com/extend/plugins/wp-hide-dashboard/

And an inline login form for your sidebar: https://www.ads-software.com/extend/plugins/sidebar-login/