Does the plugin support the creation of multiple lists, each with their own categories?

Of course, you can create multiple lists. Say, you want to cover a broad niche like Best WordPress Plugins (this is your category). You can create multiple lists for different types of plugins like Gallery, Portfolio, SEO etc. Then you add links under each Lists and assign all these all Lists to Best WordPress Plugins category. Finally you call that category with short code on any page of your site.

Here is the live example of the above scenario: http://www.dna88.com/ultimate-list-of-best-wordpress-plugins/ . Related to 1st question check out how the Search feature works. The link Lists become Filterable tabs below the search bar.