Around 1600, was Messer Grande, the chief of police, a nobleman
elected or a citizen appointed?  If the former, how long was his term
of office?

Are you referring to Messer Grande who is mentioned in the Memoirs of Casanova?

"The date was July 25th, 1755. Next morning at day-break who should
enter my room but the awful Messer-Grande."
http://romance-books.classic-literature.co.uk/memoirs-of-jacques-casanova/volume-2d-the-false-nun/ebook-page-35.asp

Historians also mention Messer Grande and the notes on my edition of
Casanova indicate that the man who arrested Casanova had held office
for 5 years or so.  That suggests he was appointed, because most
electoral positions were for limited terms (although some could be
renewed without a break).  The mere sight of Messer Grande could
reputedly break up a riot.  He also had a deputy, (Vizio).  If
appointed he would be a citizen and if elected a noble.

Well, just in case you understand Italian, "I Servizi Segreti di
Venezia" by Paolo Preto should be rather interesting:
http://www.sisde.it/sito/Rivista1.nsf/ServNavig/17
http://www.saggiatore.it/index.php?page=autore.home&id=3337

The name of the "Messer Grande" in 1755 was Mattio Varutti ("Il 26
luglio 1755, il capo degli sbirri veneziani Mattio Varutti"):
http://art.supereva.it/giacomocasanova.freeweb/page30.html?p
http://www.creatores.de/ca23blei.htm

"Messer Grande" also written as "Missier Grande" ("Quando il 26 luglio
1755 il Missier Grande"):
http://www.filiarmonici.org/casanova.html

However, my comment doesn't answer your question.