He became too tired to carry his friend on his rope and decided to cut it
I don't know how you carry someone on a rope, but the grammar in that sentence is fine.
1)Though with a broken leg he managed to arrive at the camp by moving his hands and knees OR is it OK to say with his hands and knees. Could you omit by before the gerund "moving"?
I would write, "Although he had a broken leg, he managed to arrive at the camp on his hands and knees."
2) At first he liked being Hyde but after (a while/a little period of time/after having murdered a man) he decided to stop his transformations. Are they all OK???
I'd write, "At first, he liked being Hyde, but after murdering a man, he decided to stop his transformations."
3) it is not correct using animals for tests???? OR it is not fair to use animals....
The wording for this depends on what you mean. If you mean it's against some rules or laws to use animals for tests, then your use of "correct" works. If you mean that this practice is against your sense of morals, than your use of "fair" is correct.
#1 is grammatically correct. It needs a period at the end. However, it doesn't make good sense. It would be impossible to carry someone on a rope.
#2. Put a comma after leg. Instead of using "moving", use "by crawling on his hands and knees".
#3. All three of those phrases may be used. However, you need a comma after Hyde, and you would need a comma after whichever one of those phrases you would choose to use.
Last one,I would reverse those sentences. "Using animals for test" would be the subject of the sentence.