Can I edit or add new listing fields?

Yes! You can edit the listing fields manually by tackling the PHP code or using a 3rd party plugin.

There are three main ways to customize the fields in WP Job Manager;

1. For simple text changes, using a localization file or a plugin such as the Say What Plugin. See Translating WP Job Manager for more information.
2. Use a 3rd party plugin such as WP Job Manager Field Editor which has a UI for field editing.
3. Use the WordPress hooks (filters) which are explained here.

Above is an excerpt from the WP Job Manager official documentation.