What is the ruling on downloading stories from a website that offers them for free and has "All Rights Reserved" written on them, given the uncertainty of obtaining the original story owner's consent?

The responsibility lies with the website owner who authorized the download of the stories. It is not incumbent upon a person to search for the story's owner to ascertain their permission. Referencing the source is sufficient. However, if it becomes clear that the website does not uphold academic integrity and intellectual property rights, then it is better to refrain from taking from it, in accordance with the Prophet's (peace be upon him) saying: "Leave that which makes you doubt for that which does not make you doubt."