How to further follow up after contacting project supervisor twice?
Too late to be useful, but I'm on an "unanswered questions" kick, so...

Yes, I would contact him again. The worst that can happen is that he is annoyed; the best that can happen is that you get your internship. 

How to structure it? Concisely. Like this:


  Dear Professor, Sorry to pester you, but if we move ahead with this, I will need to get the signatures/stamps from my home institute this week. Is there any update about my status? I am still highly interested in this opportunity. Thank you, -Name.