Can we mix the genes of two different species to produce a completely new species of animal?
In theory, yes--we could combine the genes of two distinct species to produce an entirely new species. In practice, this is a little more difficult. In order for an animal to be part of a legitimate species, it must be able to produce viable offspring. Some species are similar enough functionally and genetically to mate, but not produce fertile offspring. For example, horses and donkeys are functionally and genetically capable of mating to produce a mule or a hinny. However, all mules and hinnies are sterile and incapable of producing offspring, even when bred with other mules or hinnies. For this reason, mules and hinnies are considered hybrids rather than distinct, viable species.
This kind of genetic splicing really entertains the mind and has been a feature of many science fiction films, like Jurassic World. In reality, this kind of genetic engineering is far less buffet-style than it may seem. We can't just take an elephant's trait for tusks, a cow's exceptional milk output, and a frog's ability to croak, shake them up in a bottle, and get an interesting new animal. Genes which are spliced together must be compatible to produce an organism which may then create viable offspring. With species which are quite different from one another, this is pretty tricky because there are some traits we may not want to remove or cover-up. Some traits just don't work together at all. What's more, not all expressed traits are hiding in just one gene or sequence of DNA--some traits are written in many parts of the genome or all throughout it.
It might be helpful to think of this like baking! If you have recipes for dinner rolls, chocolate chip cookies, and a sponge cake, how might you combine the best of each into one super baked good? You can't just dump the ingredients for all three recipes into one bowl and expect it to come out the way you want. Even trying to add or remove one ingredient here or there to a base recipe could be disastrous. Genetic engineering is much the same way! Through careful study, experimentation, and testing, we just might be able to create new species.