Some issues (as outlined in the Federal Constitution) have been assigned to the Federal government for management - these are almost entirely things that cross state lines - Interstate Commerce, for example. We don't want teh states setting up arbitrary taxing schemes so that when some store in Oregon buys their wholesale goods from a distributor in Nevada they get hit with taxes. We also want transportation systems so that goods and people can move around between the states without things like border crossing systems. Those sorts of things fall almost entirely into the Federal realm but the States have a small part in them too. All of the Federal laws have effect on all of the states. In some cases all there are is Federal law. In other cases there are Federal laws and complimentary State level laws.
Anything that isn't mentioned in the Federal Constitution defaults to the States or to the individuals ( that's what the 10th Amendment to the Federal Constitution say anyway!
). So things like marriage laws, gambling, legal hours of operation of businesses open to the public, housing rental issues, aren't mentioned in the Federal Constitution, so they default to the States to regulate. When a State makes law it is only binding on those under the jurisdiction of the state. It doens't affect people in other states (not directly anyway) and it doesn't override Federal laws if they exist.
Recognized Native American tribes (such as this one) fall into a hazy middle ground. By definition they are a "Nation" not a State. But as condition of the Federal recognition they agree to accept Federal laws and the Federal Constitution as superior to their own laws (in return they get access to some Federal monies and benefits) so the net affect is that they are treated like a state in many ways. Marriage laws is one of the areas they the individual tribes are allowed to decide on for themselves. Tribal laws work pretty much exactly the same way State laws work.
As far as the claim made in the article... Well, it came from an anthropology professor... I'm sure he's a nice guy and all but if you read a bit further down in the article the Tribe's own lawyer pretty much disagrees with him and admits that they don't expect that it will hold up if poushed against the Federal law.
Adoption is a State/Tribal level issue. The Feds don't really get involved with it much UNLESS it involves international adoptions. They get involved there because of issues of citizenship and such. Reconginzed tribes are allowed to control the adoption process for their citizens (and that goes for both citizens and adoptees and those doing the adopting.) Most tribes have agreements setup with the different states to handling things like a Non-Native American adopting a Native American child or the adoption of a non-Native American child by a Tribal adult. (Some tribes don't allow their tribal children to be adopted by non-tribal adults at all.)