Were there any crops and livestock common to both the Old and New Worlds prior to the Columbian exchange?

The Solanum family might be an interesting contender...

In the Americas, they yielded tomatoes, potatoes and chili peppers (red hot or otherwise).

In Asia they yielded the eggplant (aubergine), but in Europe the solanaceae were best known for the Deadly Nightshade.

Which probably explains why many 16th century Europeans thought the potato was evil (3rd para) when it was first introduced. As were tomatoes...

Reindeer/caribou (Rangifer tarandus) have been herded by Arctic peoples since time immemorial. They are herded rather than farmed. The conditions in the far north require that the herds migrate, so a sedentary/farming lifestyle based on reindeer/caribou can't work, and the degree of domestication is minimal. They are used for meat, skins and labour, humans play a role in gathering, protecting and controlling the herd, and so they qualify as "livestock".

Wikipedia notes peoples in both the Eurasian and American Arctic lead, or have led, nomadic lifestyles based around the Reindeer:

Arctic peoples have depended on caribou for food, clothing, and shelter, such as the Caribou Inuit, the inland-dwelling Inuit of the Kivalliq Region in northern Canada, the Caribou Clan in Yukon, the Inupiat, the Inuvialuit, the Hän, the Northern Tutchone, and the Gwich'in (who followed the Porcupine caribou for millennia) [These are North Americans]. Hunting wild reindeer and herding of semi-domesticated reindeer are important to several Arctic and sub-Arctic peoples such as the Duhalar [Monglia] for meat, hides, antlers, milk, and transportation. The Sami people (Sápmi) [Scandinavia] have also depended on reindeer herding and fishing for centuries. In Sápmi, reindeer are used to pull a pulk,ba Nordic sled.

Barley is another example. Barley of the Old World and the Little barley of the New World are different species but the same genus, hordeum. From the abstract in

N. Mueller et al. "Growing the lost crops of eastern North America’s original agricultural system." Nature Plants, 2017, vol. 3.

Thousands of years before the maize-based agriculture practiced by many Native American societies in eastern North America at the time of contact with Europeans, there existed a unique crop system only known through archaeological evidence. There are no written or oral records of how these lost crops were cultivated, but several domesticated subspecies have been identified in the archaeological record. Growth experiments and observations of living progenitors of these crops can provide insights into the ancient agricultural system of eastern North America, the role of developmental plasticity in the process of domestication, and the creation and maintenance of diverse landraces under cultivation. In addition, experimental gardens are potent tools for public education, and can also be used to conserve remaining populations of lost crop progenitors and explore the possibility of re-domesticating these species.

Little barley is given as one of the examples of these crops (page 1 of the paper).

Dogs were common to both the Americas and the old world, since at least some of the prehistoric ancestors of American Indians brought dogs with them from Asia.

The plains Indians used travois pulled by horses to transport their goods. And before they had horses acquired from Euorpeans they used dogs to pull travois with much lighter loads.

I think that you should also try to find out whether any domesticated chicken species existed in both the old world and the new world before the Columbian exchange. I think that chickens have been used as an argument in favor of hyppothetical pre Columbian contacts.

Cotton

As far as crops are concerned, I would say that Cotton is certainly one of the best candidates. Although the species are different, in both the old and the new world, different civilizations have been cultivating cotton for several millennia. History of cotton

As reported in the article linked, Christopher Columbus, in his explorations of the Bahamas and Cuba, found natives wearing cotton, a fact that may have contributed to his incorrect belief that he had landed on the coast of India.

Beans

The beans, although of different species, were grown both in the old and in the new world and were an important source of protein throughout history.

Wild Species

in the list of foods there are also species that are not properly cultivated but that in some way were consumed both in the old and in the new world.

For example strawberries were harvested for medicinal use since Ancient Rome and, in later times, cultivated in small gardens in Europe. Charles V, France's king from 1364 to 1380, had 1,200 strawberry plants in his royal garden.

Wild rice is another one. Native Americans harvest wild rice by canoeing into a stand of plants. Several Native American cultures, such as the Ojibwa, consider wild rice to be a sacred component in their culture. ‎‎‎‎‎ ‎‎‎‎‎