Yes, the premium version of the plugin allows you to add a custom image. In Badge options, you will find the upload button and in the Opacity and position section, you can set the badge in the product image.
Yes, the premium version of the plugin allows you to apply badges to categories.
Yes, it works with ‘variable product’ types. However, there will be no DIFFERENT badges for each variation, the badge depends on the variable product (it does not change if you select a different variation).
No, unfortunately, it is not possible to assign a badge by tag. You can, however, assign a badge to all products belonging to one category. Please, refer to this page for more details.