Is there a CiviCRM extension for managing a book and video library - or some other library management system which integrates with CiviCRM so it can see the membership status out of CiviCRM (only current members are allowed to borrow from the library). Thanks in advance for your answers.
if you are using Drupal - and you aren't needing to use clever "enter the book number and have all the details pulled in from external software" when adding a new item to your library, then i could see a relatively easy path of making each item a node, using unlimited group of fields to track 'out date, in date, person'- and use civicrm entity 'contact reference field' to track which civi contact has the item
and do the rest with views and webforms
there is also https://www.drupal.org/project/library but it may be as much effort as building from scratch imo
There is no specific Library extension that I am aware off, nor is there an integration with KOHA AFAIK.
I think you would either have to develop an integration that allows koha to check if someone is a member in CiviCRM (which would be relatively straightforward with the API) or develop an extension that adds the required functionality to CiviCRM.