The question has two parts, which I will address separately.
What process can we use to create animal meanings?
Most myths and legends begin, when a group of people are sitting around campfire, in a cave, or in a hut... Outside Mother Nature unleashes her fury, by sending torrents of rain and casting roaring thunders. Wind slams against the walls, yet in the middle of the darkness you think that you saw something. A movement maybe? Or was it a pair of eyes, full of hunger and hate?
Before you can ask the guy next to you if he saw the same thing, the oldest man in the room, an experienced hunter and brave warrior, begins his tale...
Have I ever told you, how I lost my middle finger? No? Well then, you see, wolves have a very peculiar habit...
If you want to create a good myth, imagine yourself sitting in such hut, hungry and thirsty, listening to old man, looking at his scarred face, while he tries to scare the shit out of you and comfort you at the same time.
And now fast-forward 50 years. You are now the elder, perhaps you are missing a limb or an eye. All eyes are fixed on you, waiting impatiently for words of of wisdom. You take a deep breath, take a sip from your cup and start retelling the same story you've heard so long ago in a lonely hut. Sometimes you omit something, sometime you add something from yourself, but one way of the other it's still the same story about peculiar habit of the wolves.
If you want to create myths, put yourself in this mindset, and you shall not err.
What leads cultures to choose specific animals for specific meanings?
One way or the other - fear. Fear comes in different shapes and forms.
Some are afraid, that they will meet a giant bear, while hiking in the woods. So they build little shrines, so the spirits of the forest will divert the bear from their path. Others know how to use that fear - they dress up as bears to strike destroy courage of their enemies, or they hunt for the huge animals to gain respect of their tribe.
Bears are fearsome, because they are big, strong and hard to kill, and those qualities are recognized in bear-related myths.
Some are afraid, that a pack of wolves will attack their cattle, and they will watch helplessly, as their food supply is destroyed before they very eyes. So they build little shrines, to divert wolves to the next valley. Others know, how to use that fear - they mimic a pack of wolves while attacking an enemy, to make an impression of one, perfectly coordinated killing machine.
Wolves are fearsome, because they are excellent trackers, they have strong jaws and sharp claws, and they can work as a team. Those qualities are recognized in wolf-related myths.
Some are afraid, that a fox will sneak into their farm and kill all the chickens and they will starve. So they fortify their henhouses hoping to stop the animal.
Foxes are fearsome, because they are smart and agile. Those qualities are recognized in fox-related myths.
Some are afraid, that their cattle will fall victim to a disease and die, or that they will not reproduce.
Some are afraid, that rats will infest their pantry.
Some are afraid, that a certain species of bird will not make its nest on the roof, which is a sign of bad luck.
We can go on like this forever, but I think you get the idea. Think of an animal, connect some sort of fear to it and turn it into a story.