I don't know about whether or not the said "person has too much time on their hands", but if definitely sounds as though they have ulterior motives in mind , and that is to scam unsuspecting people out of their hard-earned money!
Fifth floor of a 4 storied property?!!!! Second attempt to gain a licence?!!! "the city requires notes from property owners giving permission to their tenants to use their property for short term rentals"?!!!
All of those things and more should mean that you contact both "the city" (or whoever it was that sent you those 'demands' , as well as your local Police Department, together with - possibly - whoever is responsible for local and national taxes, and let them know that the [named] person whom you don't know anything about has a premeditated plan to defraud, and that you want the said Departments to investigate (put it all in writing, and ensure you obtain Crime numbers and names of those with whom you contact).
I don't know about the word "insane" however, as most so-called 'insane' people I have known have been more sane than those who are said to be completely sane! and furthermore, such an intended and premeditated scam to defraud, could only be undertaken by those who are fully compos mentis!