In the long run, the animals whose populations grow will be those that either prove themselves valuable to humans or that prove hard to eliminate. In a resource-starved highly over-populated Earth, the choice of who survives--human or animal--is likely to be won by the human (ignoring the impact of sub-species such as attorneys).
The animals we take with us as our civilization expands into the cosmos are likely to be numerous. Those limited to a meager existence in zoos and parks can't be viewed as successful, but at least their lives will be in a rather pleasant captivity. Modern zoos are more like a Hyatt Regency than Alcatraz for their occupants.
Humans will likely keep our pets, the dogs and cats that provide us with love and companionship. Cats seem especially suited to a life in zero gravity--I have no problem imaging cats thriving in such an environment. Dogs, to me, seem to need gravity for happiness (running, jumping) but they'll adapt, I'm sure.
The other animals we take with us are those domesticated ones that taste good. We are, after all, omnivorous, and no amount of processing is likely to give an algae cake the taste and texture of a steak. I could be wrong, and there is a huge efficiency drop if we choose to eat animals instead of plants, but it seems that in a wealthy society, we'll find a way to raise cattle for meat and milk, chickens for meat and eggs, pigs for bacon and ham.
Better (more efficient) choices exist for meat animals; goats produce much more milk per pound of food consumed, rabbits much more meat. Chickens are quite efficient as-is. But you can't prepare a prime rib from rabbit. Still, these choices are likely to be early winners, in some cases because they eat different parts of the plant than we humans.
Seafood will likely be available, also. We already raise salmon, catfish, and other seafood in farms. These are likely to do quite well in space, at least as long as we can find and utilize large volumes of water (I like comets). We'll miss many foods from the top of the food chain (such as tuna, swordfish and the like), but varieties of others are likely to be plentiful, possibly even critters such as shrimp and lobster.
My question for today is, how many animals will succeed against our will (such as mice, or pigeons, or ants, or roaches)? Or what others must we bring along because they are a necessary part of the ecology? For example, must we use bees for pollination? Earthworms to churn the soil?
Here's a scary thought: What if there is some pest whose presence is necessary for long-term health, such as the mosquito? Some of them can't reproduce unless they've consumed human blood, but has any human ever reproduced before being bitten by a mosquito?