Depends, but mostly no. In most bands or tribes a person's main "job" was feeding themselves and their families. They would spend their time hunting, fishing, gathering, or (in more complex societies) farming. When they weren't doing this they were crafting tools, making pottery, caring for children, fixing clothes, building buildings, or the hundred other things that needed doing. There wasn't a group of "official hunters" that hunted all day while other people took care of other parts of the band. Work wasn't typically specialized like that. If you were feeding more than your immediate family it was because you were feeding your extended family, doing someone a favor, or participating in an activity with the entire tribe such as a seasonal acorn harvest. There was rarely someone "in charge" who could tell people to do certain things or assign duties; there were people who were respected as leaders but they had no authority so no one had to obey. Even shamans were more of a part time job they did in addition to all the other activities listed.

In chiefdoms or state-level groups, however, we see some job specialization. These were native societies that had a ruler of some description and that ruler's whole job could be "ruling" or they could rule in addition to all the other things they needed to do. For other noble families or lesser chiefs politics might be their entire job too. In some cases this ruling class was supported by taxes in the form of food. Some places, like Cahokia, may have even supported an artisan class and a priest class but these places were an extreme minority.

Note: This is an enormous generalization of various customs and regions and as much of it is drawn from scholarly conjecture as from historical and ethnographic evidence. To speak about "Native Americans" as one historical group is mostly foolish, as lifeways and culture could vary drastically in just the space of a few miles.

I've mostly drawn this information from "Seeking Our Past" by Neusius and Gross and from "This Land was Theirs" by Oswalt.

Most of my knowledge has to do with the Crow Indians. This book is a good source: http://books.google.com/books/about/The_Crow_Indians.html?id=Qwp8nvlwKkMC

I believe Dances with Wolves was supposed to be about the Sioux Indians. Both were plains tribes whose subsistence largely depended on the American bison (buffalo), so there are many similarities, but that is also the problem with talking about Indians; there were so many different native cultures that any generalizations you might make would not hold true across the board.

Like many societies, the Crow divided work based on gender. For that reason, not just anyone could "help out with different tasks around camp." Men would not be found doing women's work, and women would generally be excluded from men's spheres, especially secret sacred rituals.

Men were responsible for hunting, war (not quite the same thing as the white man's concept), and much of their time was spent on various religious activities. They would care for their horses, and they might do art-type things, make their own weapons, and religious paraphernalia. Women were in charge of butchering, preparing and preserving food, tanning buffalo hides and other hides, and making clothing. Since the white man wouldn't really consider hunting, religion, or war to be work, males basically did not do any "work" at all as the white man would define it. This probably explains, in part, why the white man thought of Indians as "lazy."

Nevertheless, there were certain more specialized jobs or positions. As mentioned, there was a chief or head man. The Crow did not have a specific name for it, and he did not have the ability to compel people to do something, but he decided when the camp would move, among other things. This position was not something you chose to do really, nor were you born into it, rather you got the position because people looked to you out of respect for your wisdom and warrior abilities.

A lot of decisions were put to a sort of vote/consensus among full-fledged warriors. In order to be part of this group, a warrior had to perform four feats, including steal a horse from the middle of an enemies camp, and take a weapon from the hands of an enemy. So, once again, it was not really something you chose to do, it was a position you were qualified for because of your abilities. Crow warriors were divided into several clubs (warrior societies). This was separate from your clan, which you were born into. A young man would choose to join one of the clubs, and almost everyone joined one of them, although there was no requirement to do so. A club would have a leader, similar to the chief. There were also honorary positions. Typically there were four young women who were honorary members who would do the cooking and stuff for the club. The Sioux Indians had a similar system of warrior societies and dancing societies, which was not explored in Dances with Wolves as far as I can remember.

Lowie describes how the head man would appoint one of the clubs to be "police." They would be responsible for making sure someone did not mess up the Buffalo hunt as well as maintaining order and dispensing a sort of justice if necessary. But, as Masterofice5 described, this was far from a full-time job, just a responsibility in addition to the other things they did. A club would take it on because of the honor involved, not because there was any kind of payment.

One position that was more job-like was the town crier. He would be responsible for going among the lodges and spreading news, sometimes speaking for the chief.

Another more specialized job would be a doctor or midwife. Lowie writes about a crow Obstetrician and his wife. At the time (the early 1900s) where were male obstetricians, although historically, only women did this job. A doctor could be viewed as the kind of job someone would choose to do; a person would be asked to doctor others if he or she had the knowledge to do so. However, there is another way to look at it: often a person would learn what plants to use as medicines in a dream or vision, so it could be seen as the spirits choosing someone to be a doctor.

I'm sorry I can't speak more toward your specific questions. You are right that (extended) family ties were very important. They did not make all the distinctions that white people make. So, for example, a maternal uncle would be considered your elder brother. This sort of thing continues to this day; what white people might consider a second or third-cousin or a great-uncle, Crows often refer to simply as "cousin." Adoption was also widely practiced, and, unlike white people, it was common to adopt adults, like to adopt someone as your brother.

As a side note, all this familial closeness could lead to some conflicts as you could imagine. And the Crows had some interesting ways of dealing with this. In a very wise arrangement (IMO), mothers in law and son in law were prohibited from interacting. Also, certain relatives (the paternal clan) were joking-relatives. They could mock each other and play practical jokes with impunity. Lowie describes how you might mock a woman for being "lazy," saying you never put up a tent, you never do beadwork or make moccasins for your husband -- so that gives you some idea of the work women were expected to do.

It is common wisdom that hunter-gather societies had more free-time than agricultural or modern societies. I wish someone more knowledgeable would speak to it, because I don't know how true it is. But whites who encountered the Indians often criticized them for having no "industry" and instead spending all their time visiting, singing, and dancing. There is probably some truth to this. A Buffalo hide would be serviceable for years, so a family would have plenty if a woman tanned one or two hides a year. Of course, there was a lot of work when there was a Buffalo hunt, and you had to preserve the meat. But once that was done, 1000 to 2000 lbs (one Buffalo) of meat would keep you going for a long time. Movies like to show people doing something, so they tend to give the impression that there was more work to be done than was probably going on in reality.

Edit: As you can imagine, killing a Buffalo was not a one-man job. The whole tribe would basically get together for the buffalo hunt, which, I vaguely recall, was shown pretty well in Dances with Wolves. So even if a person might not be a young, strong man, there was still plenty of ways he or she could help in the hunt. So the concerns about a family that lost their hunter probably comes from a misconception of what hunting really meant, at least as far as the plains tribes are concerned.