You can get all unanswered posts for a topic list by adding ?max_posts=1 to the URL in your browser’s address bar. There is a theme component that you can install on your site that adds an ‘Unanswered’ button to the site’s navigation: Unanswered Filter.
If you would like to find only topics that have not been answered by staff members, but may have been answered by other members of your community, you could add a Data Explorer query to your site that returns a list of those topics. It is now possible to allow groups to run Data Explorer queries, so you could add the query to your support group page.
A good approach for dividing support work among a team is to have members of the team watch the categories or topics that they are comfortable answering questions in. For example, I watch our #support:wordpress category, because I don’t expect any other members of the team to answer WordPress related questions.
The Assign Plugin is very useful for dividing up work among a team. We make a lot of use of it on Meta, especially in the #bug category.
A lot of the support work we do at Discourse is handled with PMs that are emailed to our support team. We use group PM inboxes to know what needs to be dealt with. PMs that have been handled get archived. We also use the Assign Plugin to assign PMs to team members. We have enabled the ‘On group messages publish group read state’ setting for our Support group. With this setting enabled, we can know which team members have read a PM.
Whispers can be useful for support work if there is a need for internal discussion before answering a question. They can also be used to let other staff members know that a support request is being handled.