I disagree with FantisticDorf on a few points, first and foremost that I don't know of any native american tribes who lived in caves.
I am Chickasaw, so I will relate to you my specific tribal knowledge - there are so many tribes and each one is unique, so a blend of styles and tropes is most likely to be what you end up with.
Many Plains tribes are nomadic, which means that they hunt bison, following their herd as they travel and migrate around, making no permanent homes, and using teepees and wigwams as shelters. I'm not familiar with a DF civilization type that can emulate this easily.
My people are known are part of a group known as southeastern mounds builders. We built mounds on top of which our cities were located. Typically most families had two simple one-room homes - a round hut that we lived in during the winters, and a square lodge that we lived i during the summers. I think that the closest DF can get to this are forest retreats, and your best bet for a race are elves.
We played a game called stick-ball that the french took and made into Lacrosse. The difference in the native american sport is that we referred to it as "the little brother of war" and we tried to resolve conflicts with neighboring tribes using a game of this. The rules were to use two sticks with small nets on the end to throw a small leather ball at a goal post, one on either end of the field. Any additional rules needed to be agreed upon before the individual game. A lot of times, one or more people would die during the game, because it is so brutal. In addition to the wartime games, we have a yearly game against our cousins, the Choctaw tribe. This yearly game continues to this day.
Might be worth pointing out that America has no native horse species. They were brought by the europeans, so depending on your period-preference, you may or may not want to include them. And while we did keep some herd animals and milk them, we did not make cheese or alcohol. Tobacco was used by tribal medicine men or Minko.
Our political leaders were elected officials known as Minko. The closest translation is governor.
FantasticDorf is spot-on with the obsidian weapons, I think. Spears, Daggers, and Axes would be the primary weapons of choice.
I think that if you are doing multiple tribes, you should probably make like two or three and be comfortable with them each being portmanteu of several different tribes, rather than trying to simulate several specific real-world tribes.