This appears to be a French location, so maybe that certificate is something more routine in that country.I've never seen #14 written out before, especially with the "authorization" part. To us, it's just common courtesy. If one of my students is going to be out of town and wants to train at another ATA school, all they have to do is tell me. I'll call the instructor of the other school and give them a heads-up that my student is coming to visit.
As far as medical stuff goes, I have a line on my membership agreement that says the student acknowledges that they are in good physical condition and recommends that they see a doctor first if they have any doubts. I don't require a permission letter from a doctor. I agree with the opinion that #16 is a CYA move, but I think it is taken to extremes.
EDIT: I see Marques got in ahead of me, with more informed information than me. Damn you, Marques.