I varied my original email including other information. I wonder if you could help me express the new things, in 4 and 7. Thanks a lot!

1) Thank you very much for your prompt reply.

2) Our students would really like to be involved in a student exchange programme with your students next year. As I wrote to you in my previous email, we are going to include about twenty sixteen-year-old high-achievers. These students are looking forward to receiving email(S) from their Irish partners.
3) You can let them write part of their emails in Italian, if you like.
4)As to students' exchanges, I'll inform you about the requirements (formalities) our students will need to fulfil in order to apply for one-year (I want to say that a student can spend a year at a foreign school and have this period recognized by their Italian school??)
5) By the way, some of our students always ask us if we can organize a summer course for them to practise their English during their long summer vacation. In particular, we may be interested in two different types of two-week courses: one for students, whose level usually ranges from low intermediate to upper intermediate; and one for those Italian teachers who will have to teach their subject in English in two years' time and usually have a low intermediate knowledge of the language.
6) I would be grateful to you if you could let me know the price for such tailor-made, two-week courses including room and board on your campus.

7) It would be extremely important for me if we could can start the email partnership next week, before the parent-teacher meeting? Can I officially inform the parents about the existing cooperation between our schools, which will ........... through various projects?

2) Our students would really like to be involved in a student exchange programme with your students next year. As I wrote you in my previous email, we are going to include about twenty sixteen-year-old high-achievers. These students are looking forward to receiving email from their Irish partners.

The word "email" is a collective noun of sorts -- no plural form except in non-standard English!

4) As to students' exchanges, I'll inform you about the requirements our students will need to fulfill in order to apply for an academic year abroad.

5) All OK

6) I would be grateful to you if you could let me know the price for such tailor-made, two-week courses, including room and board on your campus.

7) It is extremely important to me that we start the email partnership next week, before the parent-teacher meeting. Can I officially inform the parents about the existing cooperation between our schools, which will ........... through various projects?
(I'm not sure what you want to put in the blank space.)

To express the new things in sentences 4 and 7, you can say:

4) Regarding the student exchange program, I will provide you with the necessary requirements that our students need to meet in order to apply for a one-year exchange program. This program allows students to spend a year at a foreign school and have their studies recognized by their Italian school.

7) It would be highly beneficial if we could commence the email partnership next week, prior to the parent-teacher meeting. Can I officially inform the parents about our schools' collaboration, which will involve various projects?