Is it correct to say

He became too tired to carry his friend on his rope and decided to cut it
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"?
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???
3) it is not correct using animals for tests???? OR it is not fair to use animals....
Thanks for your prompt reply

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.

1) Yes, it is correct to say "Though with a broken leg he managed to arrive at the camp by moving his hands and knees." In this sentence, "by" is used to indicate the means or method by which he accomplished the action of moving. It can be omitted if you rephrase the sentence as "Though with a broken leg he managed to arrive at the camp moving his hands and knees."

2) All three options are grammatically correct, but they convey slightly different meanings:
- "At first he liked being Hyde, but after a while" suggests a longer period of time during which he enjoyed being Hyde before eventually deciding to stop his transformations.
- "At first he liked being Hyde, but after a little period of time" conveys a shorter period of time or a brief amount of time before he decided to stop his transformations.
- "At first he liked being Hyde, but after having murdered a man" links his decision to stop his transformations specifically to the act of murder. This option implies that his experience of taking another person's life influenced his decision to stop being Hyde.

Choose the option that best fits the intended meaning of the sentence.

3) Both options are grammatically correct, but they convey slightly different implications:
- "It is not correct to use animals for tests" suggests that using animals for testing purposes is morally or ethically incorrect.
- "It is not fair to use animals for tests" implies that using animals for testing purposes is unfair or unjust.

Both statements express the perspective that using animals for tests is wrong in some way, but the choice between "correct" and "fair" depends on the specific nuance or argument being made.