The office of personell affairs (人事局、じんじきょく) would be the place to ask. As for your first question, that's also something I would like to know. Do you have one university where you would feel comfortable asking 人事 about that? Otherwise you'd have to ask at immigrations yourself.
Thanks. I'm going to immigration tomorrow with the first 3 pages of the application form and I'll ask them what other documents I need.