1. (This sentence means mutual assistance) can be translated as: give a peach in return for a plum/you help me, I help you.

2. (Providing) higher education grants/student loans is an effective way to attract students to apply for colleges and universities. (If you have money to go to school, you will naturally feel at ease to apply for it. If you have no money, you may lose a lot of students and enter the society directly.) I think it can be translated better.

3. He shook hands with her in an extremely coping manner. limp means that the shaking hands are weak, and cold directly shows that he is not attentive and unfriendly at all. To sum up, he did not want to shake hands with her for some reasons, but he still had to shake hands, so he just shook hands in an extremely coping manner.