Could you please check the following sentences for me? Thank you.

1) She delayed taking a shower.
2) She was caught cheating (by a teacher).
3) Sha saw him chatting up a girl.
He was seen to chat up a girl.
4) An animal's reaction to a drug is different from that of a human.
5) I'm opposed to animal testing/to testing drugs/to carrying out experiments on animals (rather than on humans?).
6) The sign forbids fishing in the river.
They encouraged studying hard.
No-one is going to join the drama course.

All is fine except in the last sentence.

"No one" is two words, without a hyphen.

Well ... I think "join" isn't the right word to use with "course" ... it would be correct to say "... to enroll in the drama course."

Sure! Here are the checked sentences:

1) She delayed taking a shower. - This sentence is grammatically correct.
2) She was caught cheating (by a teacher). - This sentence is also grammatically correct. However, you can add "by a teacher" to clarify who caught her cheating.
3) She saw him chatting up a girl. - This sentence is grammatically correct. However, the second sentence "He was seen to chat up a girl" sounds a bit awkward. You can revise it to "He was seen chatting up a girl."
4) An animal's reaction to a drug is different from that of a human. - This sentence is grammatically correct.
5) I'm opposed to animal testing/drug testing/conducting experiments on animals instead of humans. - These sentences are grammatically correct. You can choose the alternative that best fits your intended meaning.
6) The sign forbids fishing in the river. - This sentence is grammatically correct.
7) They encouraged studying hard. - This sentence is grammatically correct.
8) No one is going to join the drama course. - This sentence is grammatically correct.

All the sentences are fine, with just a minor adjustment to the sentence 3. Everything else is grammatically correct.