Most Norse would be very familiar with the typical animals we associate with farming in Northern temperate climates.

Cats would certainly be kept around farms and housing. They are good for pest control.

Chickens are a staple that most houses would have.

Cows provide dairy. Cheese is a great way to store milk for later consumption. Since cows require a lot more resources to care for, they will not usually be owned by poor farmers. It might take years to save up enough money to get one.

Dogs would be at least as common as they are today. Not only are they great alarm systems, but their mere presence tends to keep predators away.

Goats are another great source of dairy. They also eat all kinds of things, acting as waste disposals. Most farms would have a few goats.

Horses would be somewhat rare and owned by those with more wealth.

Pigs are another waste disposal and meat supply. Bacon and sausage are staples of every household.

Sheep provide wool and meat. Most farms would have some, along with simple tools to process the wool.

Badgers

Bats

Bears

Boars

Deer

Eagles

Elk

Foxes

Frogs

Hawks

Lynx

Mountain Lions/Puma/Panther

Owls

Rabbits

Rats

Ravens - Hugin and Munin

Snakes

Spiders

Squirrels

Turtles

Weasels

Wolves & Wargs

Dragons / Wyrms

Draugr & Gjenganger

Fafnir - a dwarf that was transformed into a dragon by his greed.

Gullinbursti - a golden boar made by the dwarves.

Hafgufa / Lyngbakr

Hellhounds / Garm - Garm is the guardian of Hel and the largest of the hellhounds.

Kraken

Landvættir

Marmenill (Mermen)

Nixie

Sea Serpent

Trolls

A lot of what we “know” about the races come from more modern interpretations and fictionalization of the old myths. Those more modern interpretations got integrated into D&D from the beginning, so we pretty much just have to live with it.

There is nothing in the original sources to indicate that the dwarves were short or stout. They might have been just like humans, and current beliefs about their stature might be due to Christian influence. They were usually skilled craftsmen though.

Nothing in the original sources tells us that elves, either light or dark elves, are any different than regular humans. They are definitely not fey creatures. However, for D&D purposes we can include the pointed ears and at least some sort of fey-ness.

The Aesir and Vanir seem to basically be immortal or at least very long-lived super-powerful humanoids.

There is also nothing in the original sources that says that Jotuns are any different than the Aesir or Vanir. They look just like us humans. But for purposes of D&D, we can accept that they are larger and giant-like. In most D&D worlds Jotuns are giants, so we will go with that and have the various giant races exist under the term Jotun.

This race is fairly new to D&D and isn't part of the Norse history or mythology. For D&D purposes we can say that they are a mortal child-race of the Jotuns. So if Stone Giants, AKA Jotunstein, exist then there should be a sub-race of goliaths that have similar traits to the stone giants.

These constructed humanoids are produced by advanced artificers and wizards, so they are usually found only in the largest of cities. For our purposes they would have been built in Constantinople to serve as guards or in the military. Once their twenty years of service is up they have earned their freedom, so they can then become adventurers. I have a special homebrew background I created called Varangian that players should choose for their warforged characters.

These cat-like creatures usually come from around Egypt. They are natural enemies of Gnolls, who typically can be found in desert areas.