Can I say,

(a) The next morning, the hare and tortoise started to race.

(b) In the race, the hare ran fast. The tortoise crawled far behind him.

(c) When the race started, the hare ran fast. The tortoise crawled behind the hare far away.
B. is okay, but it might be clearer if you said something like: "The tortise crawled along far behind him."

C. is incorrect.
"Far behind" means that two are on a similar path but that one is noticably behind the other. Think of a horse race on a track. Movement is involved.

"Far away" means that one object is distant from another (the direction does not matter and there is no similar path). Think of two cities on opposite sides of a nation. No movement is involved.

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This is a famous fable.
Can I only use "tortoise" and "hare"?
Can "turtle" and "rabbit" be used in this story when you tell your kids?
I would use article for both: the tortoise and the hare.
As to tortoise vs turtle and hare vs rabbit, there are some technical difference in the terms. Tortoise, for example, is a subcategory of turtle and refers only land turtles (sea turtles are not called tortoises). However, for storytelling, you could use the more common words "turtle" and "rabbit."
Thanks very much!
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