can I hide categories?

You can set the limit of displayed links to 0
this way the category wont show in the sidebar.

It would be nice to limit based on login status.

Might look at Category Access at the bottom of this page:
https://www.coppit.org/code/

Already have that installed. Works fine.

But the sidebar listings still shows the categories existing even if you can’t view the contents of them.

ie: I wouldn’t want Bob to see a category named “Things I hate about Bob” even if he can’t see or read the posts in that category.

Wonder if you are going to have to use the Role Manager Plugin, create your own capabilities (and roles, then in your sidebar.php (or where ever), test for

<?php if ( current_user_can('see_derogatory_categories') ) : ?>