Does the specialisation "hermetic" count towards the minimum of 5 for writing a book?
I would say so. If you are writing about Arts or hermetic skills (Parma, penetation or finesse). if you are writing about something else I would say no.
Yeah, that's a judgement call, but imo... sure.
What a troupe/SG should weigh here is whether the "specialization" is abusive, too all-encompassing or undefined, whether it just doesn't make sense, or just doesn't feel right. "Hermetic usage" is, imo, none of the above, but that's my saga, not yours. 
You could argue that it "doesn't make sense" - that either one has the mastery and vocabulary to write, or they don't, but imo that's a thin quibble, and not worth the headache. A good SG has gotta give the occasional gift to players - this is one that seems like it might be a good battle to choose to lose.
Yes, of course. Specializations raise the skill level by one in their subject.