I want to obtain permission to reproduce a copyright-protected image of the characters of a television program for a personal product. I am aware of the companies that own the rights to the characters, but I have no way of verifying the original creator of the specific image I consider using. I imagine such information would be necessary if I intend to reproduce the image, correct? Would I need permission from a licence holder, or does the copyright holder have the right to grant permission regardless?

I would get written permission from the property owner to reproduce the original work. Check with the US Patent Trademark Office for the owner of a work. You can also check with Better Who Is for the owner of a domain name.

You can not just take someones property. If you do you risk civil and criminal charges. Dont be lazy and say you can not find that person. Thats no excuse. If everyone just took another's property than no one would want to develop ideas and no one would have any property rights. Thats what make this country so great is the protection of property. It encourages entrepreneurial behavior.

Whether you can get permission for additional reproduction would depend on the license agreement currently in place, it may be an exclusive license for a term of years and the original holder may not have the right to grant anyone else a license.