Extra fields on categories that is available via the WP-API?

The best way is to use Advanced Custom Field plugin. It takes away all the hassle to create the fields, and update them.

Procedure could be:
1. Install “Advanced Custom Fields” plugin
2. Create a new field-group
3. Add desired type of field(s)
4. Add this field group to category

Now, you can see your added field(s) on the “Add New Category” or “Edit Category” pages.

Then you can just fetch category object(s) and use this code block to get your field value:

 $terms = get_terms( 'category' );
 if( !empty($terms) ) :
    foreach($terms as $term) {
        $fieldValue = get_field('field_id', $term );
     }
 endif;

This is to get all categories. You can simply use it for a single cat object.

If you’re willing to get your hands dirty with the coding, try this. http://en.bainternet.info/wordpress-category-extra-fields/