Nancy,
There are communities in Alaska that are 100% tribal members. Maybe you folks don't have them up there but we do down here. I am pretty sure there are villages up North where the majority are Tribal members. Tribes that have tribal court can prosecute tribal members for low level crimes. Non-Natives can elect to be tried in tribal court for a crime or in a state court. You might be surprised that many of the tribes in the lower 48 have laws almost identical to our state's law. They have to follow federal law, that means people have the same rights under the United States Constitution when it comes to being arrested or convicted for a crime.
As for child protection cases, they will follow ICWA as it is a federal law, regarding placement of children removed from their homes and follow it for adoptions.
If a Tribal member is arrested in Anchorage for a crime, they will still be prosecuted in state court. State laws have to be obeyed as do Federal laws.