Why can I build barracks in some homeland provinces, but not in others? Am I missing something?
If you are in the Camillan period (indicated by a "building" in your cities), then you can only build roman barracks in your original cities + Segesta (I think). When you get to Polybian Era, you'll be able to build those barracks in Italy. During Marian period, you can build roman barracks anywhere with a Type I or II government.

How long does unrest last in type I government cities?
I don't think the government type has any influence (besides the boni it gives) on unrest.

Hope this helps!