Could you please check these statements, please? Thank you.

1) the first who trusts/to trust are both possible?
2) "Let's check the answer to those exercises"
She proposed checking the answers...
She told/ordered us to check....
She suggested checking the answers
She apologized to me for keeping me waiting.
3) Cheetahs have always been poached by poachers (??)
4) At first he didn't risk coming into the camp, but after 20 minutes he dared to come (into it??) and began feedin on the fruits.
5) The elephant represents hope because he has learned how to trust humans.
African cheetahs capture their prey in semi-arid desert regions.
6) He was the first that (correction: who/to) learnt to trust humans by going near the camp to eat food.

1) the first who trusts/to trust are both possible?

I can't tell without seeing the phrase in a complete sentence.

2) "Let's check the answer to those exercises"
She proposed checking the answers...
She told/ordered us to check....
She suggested checking the answers
She apologized to me for keeping me waiting.
All are fine.

3) Cheetahs have always been poached by poachers (??)
It's redundant. I'd say this: Many cheetahs have been killed by poachers.

4) At first, he wouldn't risk coming into the camp, but after 20 minutes he dared to come in and began feeding on the fruit.
See all the corrections?

5) OK

6) He was the first to learn to trust humans by going near the camp to eat.
See the corrections?

1) Both "the first who trusts" and "the first to trust" are grammatically correct constructions. However, the choice between the two depends on the context and intended meaning. "The first who trusts" implies that there are multiple individuals who could potentially trust, and the first one who does so is being described. On the other hand, "the first to trust" implies that trust is an action or decision, and the first person or entity to make that decision is being described.

2) Let's check the answer to those exercises: This sentence suggests that the speaker and the listener(s) will be checking the answer(s) together.

a) She proposed checking the answers: This implies that she made a suggestion or put forward the idea of checking the answers.

b) She told/ordered us to check the answers: "Told" indicates that she gave a directive or instruction, while "ordered" suggests a more authoritative or forceful command.

c) She suggested checking the answers: Similar to "proposed," this implies that she made a suggestion but without as much emphasis or authority.

d) She apologized to me for keeping me waiting: This implies that she acknowledged that she caused a delay or made you wait and expressed regret for it.

3) The statement "Cheetahs have always been poached by poachers" is correct grammatically. However, it is important to note that "poached by poachers" is redundant because poaching specifically refers to illegal hunting, so the word "poachers" is already assumed. Therefore, you can simply say "Cheetahs have always been poached."

4) The corrected statement is: "At first, he didn't risk coming into the camp, but after 20 minutes, he dared to come and began feeding on the fruits." The phrase "into it" is not necessary because the meaning is clear without it.

5) The statement "The elephant represents hope because he has learned how to trust humans" is grammatically correct and conveys a clear meaning. Additionally, the statement "African cheetahs capture their prey in semi-arid desert regions" is also grammatically correct.

6) The corrected statement is: "He was the first who learned to trust humans by going near the camp to eat food." In this case, "who" should be used instead of "that" to refer to a person.