5. The woman's problem is that she does not want to give up her apartment when she goes to the Italy for exchange program.
The man suggests two possible solutions. The firs suggestion is to give up the apartment and ask her parents for help about finding a new place just before she returns from Italy. The second suggestion is to swap her apartment with the apartment of a exchange student from Italy.
I think the first suggestion is better for her. This is because the university provides free accomodations for exchange students. So it would be hard to find a person who wants to receive woman's proposal. (What is that?) Also her parents are willing to help her out about finding a new place. Even if the woman feels burdensome about asking for help, parents would not care about it.
For these reasons, I would recommend the first option for her.
* 국가 명칭 앞에는 한정사를 쓰지 않으므로, 앞에 the는 생략해주셔야 합니다.
woman's proposal이 exchange students하는 제안이라는 점을 설명해주세요.