Yes, by adding a custom field to the taxonomy with the type image. Locations by default include this image field, other taxonomies do not.
Is it possible to show excerpts of each listing on the category pages?
Yes. Using the Backend Display Editor, add the field (Listing Description) to the summary display of the listing. See the documentation on customizing displays.
Can I choose custom icons for categories or am I restricted to those you include with Directories Pro?
Currently, this is limited.
How many levels can categories have?
Can I use categories associated with another post type, such as blog posts, for my directory?
This is not currently possible.
Is there a limit to the number of terms one can have associated with a listing?
Currently term data is cached up to 1000 entries which may cause some terms not to appear. One can alter this limit manually in /lib/components/Entity/Helper/TaxonomyTermsHelper.php
Edit line 25:
->fetch(empty($num) ? 1000 : $num);
They plan to add a hook for this to allow configuration from the backend in the future.