Vegetarian diets are a good way to kill a cat. You can keep them alive with supplements but it's never going to work long-term. They are
obligate carnivores which means they
need meat. It's cruel and should be considered animal abuse IMO.
Dogs are a bit more adaptable. They're classified as carnivores (this shouldn't be up for debate...the only people who tend to say otherwise are some pet food companies selling plant-based foods and anti-raw folks), but not obligate carnivores. In other words, they can consume some plant-based foods but are still designed for eating meat. If you need proof, look into the digestive tract (short and extremely acidic), teeth (see any molars for grinding plants? and notice the sharp canines), jaws (made for crushing, it's not capable of the motion required for chewing like we do), etc.
A dog will probably do better on a homemade vegetarian diet than some of the worst kibbles because they're at least getting fresh foods. But neither will ever be anywhere near optimal or natural. I'm vegetarian but would never feed my dogs as vegetarians, it's not fair to force your own views on them as far as diet goes. If someone wants a vegetarian pet they should buy an herbivore. Rabbits are awesome for that.
From what I've read on home diets; meat should include organs and all other things that humans don't typically consume when we eat our meat. I've read supplements are only needed if you're feeding the dog(s)/cat(s) muscle meat exclusive of any organs.
In any case, as far as dogs go I'd look at what foxes eat in their environment. And I say this because several months ago I watched an interesting NatGeo show called "Dogs Decoded" in which they reveal our dogs true ancestors are likely foxes, not wolves. So I guess feeding a dog a vegetarian diet, I'd feed them what a fox would eat, just exclusive of any meat sources.
Also I think it depends largely on the size of the dog. I would NEVER feed a large breed dog a vegetarian diet, but a small dog could probably do quite well on one if that's what the owner wished.
We should be consuming organs if we eat meat, it's just not common in the American diet these days. But yes, organs should be part of a homemade diet. Even supplement-heavy BARF and homecooked diets tend to at least include liver (it's the most important organ IMO and the one wolves consume first).
I watched that special and I have no idea what you watched because it never said that. Dogs are domesticated wolves and it's not even up for debate now since leaps were made in genetic research. They are
definitely not foxes.
Why would small dogs be any different diet-wise? They're still carnivores and should be eating meat.
I don't remember if the show covered chromosomes, but what they did find, is under domestication from humans, I think around the third bred line foxes were born baring color patterns similar to our dogs today, their tails began curling over their backs and they started looking like your common Spitz breeds. Meanwhile, domesticated wolf pups didn't physically change in any way. Plus, in comparison to wolf puppies, kits acted exactly the same as the domestic puppy in maintaining eye contact and coming closer when called, whereas wolf puppies never directly looked at humans, avoided the humans when called, and were unruly no matter how many lines were bred and domesticated. You might find it interesting to watch, too.
Anyway, I personally wouldn't feed a large dog a vegetarian diet based on sheer size. I think it would be much more convenient to feed a smaller dog a vegetarian diet, were I to do it, because they don't consume nearly as much as a larger breed.
I don't think you interpreted the special correctly. This is something I've studied extensively and taken classes on (although my degree is just in general Biology I have a huge interest in Genetics and Evolution). The fox experiment showed that there are certain genes tied to traits we tend to select for in domesticating animals. When we domesticate animals we tend to select for (sometimes unknowingly) the more docile, friendly, biddable animals. These traits are genetically tied to traits such as floppy ears, various color patterns (piebald being a famous one!), and curled tails. You'll notice those same types of traits in sheep, pigs, guinea pigs, rabbits (that's how we got lops!), etc. Cats are the exception because we never enforced any breeding selection on them (they tended to run loose, often as barn cats, and bred without our interference). Now, though, we're getting cat breeds with floppy ears, bobbed/curled tails, etc. just like other domestics. It's actually really fascinating and I could dig up some articles and books for you if you want

.
The wolf experiment they talked about proved that domestication is an example of nature over nurture. Wolves can't be domesticated simply by raising them as pups, it's all genetics. The wolf experiment would have resulted in dog-like wolves, just like the fox one, if it had been done on the same scale and for the same amount of time (remember that the fox experiment was very, very long and those foxes are still used today...no such experiment was done on wolves).
The guy who did the fox experiment actually used other animals, too, people just tend to forget/not care. He got similar (just not as dramatic) results with his rats, otters (I forget which species), etc.