Yes, the plugin allows uploading a custom image as explained here.
Yes, the premium version of the plugin allows you to create rules to apply badges to specific categories as described here.
Yes, the plugin supports variable products and allows applying different badges for each variation as described here.
You can add as many badges as you want to the same product.
Yes, you can create badges to show on products with specific tags. You can find more details in this section of the documentation.
If you are already using YITH Badge Management, when updating to version 2.0, all the badges you have created will be automatically created in background. This action requires a few minutes after which your badges will show in the tab Badges and Badge rules. During the creation of the rules, the list will be empty but you will be able to create a new rule if needed.