Yes, they did!

Native Americans adopted a number of European domesticated animals. The ways in which they did so varied quite a bit, depending on the specific community of Native Americans and the animals in question. In some cases, the adoption of European livestock radically transformed Native American societies. In other cases, Native Americans selectively integrated European livestock into their societies in ways that fit into pre-existing patterns and practices. I'm not an expert in this area, but it has come up in other research I have done.

Prior to European contact, there were a very limited number of domesticated animals in North America. Dogs were widely domesticated, and many communities had "village dogs" that guarded villages and assisted in hunts. In the southwest and modern-day Mexico, Native Americans kept turkeys, and some Algonquin-speaking groups on the Eastern Seaboard kept semi-domesticated hawks around their fields to keep other birds away from their crops. In South America you also have domesticated Llamas, of course, but I'm focusing on North America for now.

In the Eastern Woodlands, where the English established their colonies, Native Americans began to raise European livestock in the 17th century, relatively shortly after contact. The most popular new animal among Native Americans was the pig. This is probably because the pig was seen as analogous to the familiar village dog–it could be fed on food scraps and largely took care of itself. Cattle were less popular among Native Americans. Cattle were large, and required far more food than pigs. Native Americans on the eastern woodlands did not have a tradition of clearing or fencing pastures, or cutting and curing hay, which was necessary to keep cattle fed during the long winters. Compared with keeping pigs, keeping cattle required a greater reconfiguration of Native American agriculture, and most groups just decided it wasn't worth the trouble. English cattle were also constantly escaping from colonial settlements and wrecking Native American corn fields, which gave them a bad name among many Native Americans. That said, Native Americans in the area did raise cattle and horses, although this remained a sideline, rather than a major part of their agricultural system.

Further west, the situation was very different. Horses, as you point out, were quickly adopted by a number of tribes in the southwest and great plains. The other classic example of Native Americans adopting European livestock is the Navajo-Churro sheep. The Navajo (Dine) acquired sheep from Spanish sometime in the 16th century and by the 18th century keeping sheep became a major part of their life and culture. This is a fascinating story, and one that I hope people with more expertise can weigh in on.

For information about the politics of animal domestication in the 17th-century North America, I highly recommend Creatures of Empire, by Virginia deJohn Anderson.

For information about sheep among the Navajo, I would recommend "The Origins of Navajo Pastoralism" by Marsha Weisiger in Journal of the Southwest, summer 2004.

Since you already got an answer from North America, I thought I'd step in to give an answer from South America. Here in Peru several archaeological remains show signs of constant use of llamas and alpacas, which were basically domesticated as a source of wool and for the transportation of resources. Llamas are not strong enough to withstand a human rider, but they seem to have been used to carry small sacks and perhaps a certain amount of products. Another important use for llamas is the production of meat, consumption of llama and alpaca was not uncommon, though most animals would still be used for production of wool rather than killed for the meat.

Another very important animal that can easily be domesticated, being both easy and quick to breed, are guinea pigs, there are signs of them being domesticated as eary 3600 BCE. They were mostly used for consumption, and still are, they are actually very VERY tasty, but also contain a good amount of fat, which was very sought for in the cold andean regions. However it seems that guinea pigs were mostly raised at home by the farmers themselves, while there are clear signs of their domestication, there are no signs of massive farmlands dedicated to them, in contrast to llamas and alpacas.

Finally, one important aspect of andean religion was the use of animals for sacrifice, various sorts of animals were used for sacrificial purposes, and as part of the ritual preparation of the animal for sacrifice, they were raised at temples, not caught in the wild, and were taken under the protection of shamans, well fed and decorated in ornamets prior to the sacrifice. These animals were not part of a massive production, like that of llamas, nor a domestic production like that of guinea pigs, but rather selected and bred for a specific ritual sacrifice, and the animal itself could vary from ducks and quails and other species that are normally not domesticated.