Good eyes, but no. Рассказ is something that you can рассказать, which comes from сказать and the prefix рас- (can take the form of раз-, разо-, роз-, etc). It is probably not related to раз "time", here it is used to add the sense of distributing, spreading outwards, cf. раздать, разбросать.
Yes, except it's этот, be careful of the typo. Russian Е is pronounced "yeh." Russian word order is flexible unlike English, but the last word in a Russian sentence usually takes the main attention. Your sentence is drawing attention to what Tom wrote. The sentence in the exercise is drawing attention to who wrote this short story.
This section of the Russian course is trying to teach noun cases of objects, and the flexible word order in Russian as a result. The subject in the exercise is "Том," and the object in accusative case is рассказ, in this case accusative = nominative because рассказ is an inanimate noun.
Your sentence translates as "Этот рассказ написан Томом." The subject in this sentence is "рассказ," which matches the subject in English in your sentence - "story," and Tom is the object in instrumental case - "Томом."
"Tom" is a bad example because it's an English name, but if the short story wrote him, then he would be in the accusative noun case.
Этот рассказ написал Тома. It's a little easier to understand using the Russian name Ваня.
Этот рассказ написал Ваню.
It's like the difference between "he" and "him" in English. "He wrote this short story," or "this short story wrote him."
OK. Then I'm back to: the exercise is bad, because it fails to illustrate how the grammar works.
Another thought: a literary phrase we sometimes see: A short story, he wrote. With the pronoun, it's clear that "he" is the subject. But the comma helps, too, as with "A short story, Tom wrote." It's weird, but if the comma is fully pronounced, it's clear that it means "Tom wrote a short story", not "A short story that was written by Tom". But this is very delicate stuff.
Although S-V-O is the most standard word order in Russian, any of the six combinations are also accepted. This paper gives more detail on the phenomenon, both giving more information and drawing psycholinguistic conclusions: http://ir.uiowa.edu/cgi/viewcontent.cgi?article=1350context=etd