Is there any way a catdog (like from the television program Catdog) could survive? If it could survive could it evolve? Are there any reasons it couldn't?
I assume your first question is if a cat-dog could be created. To that, I would say all it really takes is to cut off the back legs of a cat and a dog, attach some prostheses to aid in waste disposal, and then tape the two animals together. There would probably be some problems with even this basic approach:
1- Rejection. A cat and a dog do not mix. It's possible that even being so close to one another may cause some sort of allergic reaction on one or both of the animals. Their butts would probably get infected by the unnatural configuration/chafing, and once that happened both animals would probably die without constant medical help. This problem would be compounded if the animals were surgically attached, as each of their immune systems would work to destroy the other.
2- Balance. Even if you could get both animals trained enough to walk around without falling over or hitting into things, their midsection is going to be dangerously unsupported. For a real-life example, many ferret owners are aware of the high probability their little rodent has of developing spinal problems during its lifetime. Since a cat-dog is larger than a ferret, due to the square-cube law the spinal stresses will be compounded. Eventually, one or both animals will break their back, then the middle of them will probably start dragging on the ground, then it'll get infected, then yet again the animal is dead.
3 - Fighting. Cats and dogs don't get along. Especially if their butts are getting infected and their backs are being broken and someone keeps on pulling them backwards. These poor creatures will be scared and confused, and all hope of cooperation will be lost in the desperate struggle to survive.
Now, as to whether or not such an animal could evolve on its own, it's somewhat plausible; one possibility I can think of is a species that evolves to always produce conjoined twins. The problem here is that there is no good reason for such an animal to develop; a cat-dog is not any better at hunting than a cat or a dog, and I can't even begin to imagine how hard it would be for them to produce offspring of any kind. There are some very simple organisms that can get away with this; I seem to recall there are some species of worm whose fronts match their backs. However, for a creature that has to run, jump, and do all of the complex things a cat or dog can do, there is no benefit to this kind of symmetry.
In conclusion, no, I don't think it can happen, or even could happen. You might be able to get something that reminds people of a cat-dog, but the closer you get to the real thing, the more implausible it becomes.
There is only one problem I can see and that is the two heads. You could say that one of the heads is nothing more than a glorified tail that confuses predators. the mouth of that head could work as the anus. This is surprisingly easy to solve. It is possible but unrealistic
Let's start with something easier - the catcat, with a cat head at both ends. In order for this to evolve, that would have to be somehow beneficial to it. Now, a cat's a predator, so why would two heads on opposite sides be beneficial in hunting? Well, a predator would certainly find it useful to be able to see behind it, and to be able to reverse direction instantly. So let's work with that - in the distant past, this catcat was a large predator, which evolved bidirectional knee joints to improve maneuverability. It also, as a genetic quirk, developed a higher-than-usual likelihood of being born as conjoined twins, in this particular arrangement. Now, the occasional such twin would be a more effective hunter, and so would tend to succeed.
The only issue I can see is this: Having two separate brains, one at each end, would almost invariably be detrimental - too complicated to coordinate. It would be more beneficial to have both heads wired to a central brain. So why did our catcat never develop that wiring?
Let's take a cue from the dinosaurs. The long-necked dinosaurs - like the Apatosaurus and the now-discredited Brontosaurus - were so large that nerve signals couldn't travel from tail to head fast enough to be useful, so they developed an auxiliary brain in the hip. Now, this was barely more than a nerve cluster, just enough to handle pain reactions, but it's reasonable to suppose that a predator might need something more complicated there. So let's make our prehistoric catcat huge, comparable to a blue whale in size. Nerve signals traveling from head to head would take seconds rather than milliseconds, and a single central brain would be infeasible. So the catcat would develop two separate brains, one in each head.
Now, to support this monster catcat, we need one heck of an ecosystem - a predator that large would need huge prey in order to keep it fed. So our Brontosaurus-sized catcat needs Stegosaurus-sized mice running around, or something similar.
Okay, we've got a catcat - but it's like a billion times bigger than we wanted. Let's say a catastrophe wipes out the big prey - like the meteor that killed the dinosaurs - forcing the catcat to shrink. It retains the two-brain structure as a vestigial trait, now that it's evolved other systems around it. Once the catcat is about the size of a sabertooth, humans do the rest of the work, domesticating it and breeding it for a more manageable size. Presto! Catcat.
But you didn't ask about catcats, you asked about catdogs. That's easy enough to work in at any point in the evolution. Maybe the giant prehistoric catcat finds it useful to specialize its heads slightly differently, making the creature more flexible as a predator. Or maybe humans decide asymmetry is preferable, and start selecting for catcats that have one head that looks more like a dog.