I want to show him my international collaborative research project letter to him. Is it the right time to show him my letter after the skype interview.

I'm not sure what this letter is. If it's something you need to share with him then it shouldn't be a problem to do so?

Theoritically yes. I have done it twice when asked to provide my Master thesis during the interview. But if he/she did not ask you for further documents, I think it would indicate a sign of weakness. In my opinion, their decision will not change based on your document if they have not asked particular for it.