It's been established to me that biologists can, technically, recreate animals or bacteria from only a description of DNA. (By manually sequencing genes, which I'm sure is more difficult than it sounds.) This means, among other things, that we can recreate extinct animals, from taxidermied models, or even possibly from bones. Given enough wildlife preserves and biology labs, "extinct" and "endangered" would be things of the past.
A second possibility is that we create an animal that never has been. We would need greater knowledge of protein-folding, proteins in general, and DNA. We would also need some sort of artificial-womb technology, as this new animal wouldn't be guarenteed to gestate properly in any animal that currently exists. (I've heard we have this, but I've learned only innuendo that it exists, with no technical details, which makes me dismiss such claims.)
We need protein knowledge because DNA does not code traits, but proteins. Proteins and their foldings gave you and me the shape of our bodies, our hands, face, and eyes, and establishes every difference between you and me, us and monkeys, monkeys and dogs, and dogs and bananas. Different only by DNA, and therefore protein.
We would also need a seriously dedicated team of biologists. Biologists artificially sequencing up a bacteria was major news -- even though it was a copy of a bacteria that exists naturally. This would not only need that level of sequencing, but would also be coded from scratch. A defective code would produce no useful results, and we'd be programming with a language that we don't yet understand. A difficult proposition.
But in theory, we could have any animal we want. A chicken with 8 sets of legs, so drumsticks are plentiful? Sure. A draft animal stronger than a tank, but eats grass instead of gasoline? We can. An ant sized animal that can reproduce human speech, used as an espionage device? It'd be hard to train, but I'm confident it could be done. Farm animals with strong disease immunity? Goodbye antibiotics! An ultimate pet with the traits of all the best companion animals? Can do.
Only a mountain, no, a planet, no, a solar system of work is in our way. And some of that can be automated away.