"I have read the story" should actually be sufficient. When someone asks you "have you read this story" and you answer "yes, I have", it's implied that you didn't just read passages of the story, but rather the whole thing.
If your goal is to emphasize the fact that you read every word of the story, I'd say one of:
- I've finished the story.
- I've read the entire story.
- I've read the story to the end. (closest to your option 3)
All of your options are technically grammatically correct and mean the same thing, but they're different from the point of view of whether they'd sound natural to a native speaker. The ones with "ending" can be a little bit confusing, because the ending of a story typically refers to what happens at the end (e.g. the last few paragraphs or sentences), not the final printed character of the story. So if you read "till the ending", it may mean you haven't read the ending yet. I'm pretty sure people will understand what you mean, but it's an example that just because something is technically grammatical, doesn't mean it's what people would say.
"To the end" is a bit more idiomatic than "to its end".