Furry Writing, Worldbuilding, and Storytime

So does this pretty much mean genocide of predators species occurs? Is it more of a survival thing why some predator species join the Trophist side?

It's an unfortunate result of neither the Unitist nor Trophist sides being willing to compromise on something so fundamental, but whether it's a "genocide" depends on your specific definition. For one, Unitist revolutionary wars happen within a species or taxon, so the genetic component to predation applies to all participants in the war. It's not herbivores killing carnivores as "revenge," it's carnivores looking to move past their predator status killing carnivores of the same species who are vehemently trying to hold them back. It's kind of like how in the American Revolution, Americans killed a lot of British people, even though Americans are basically just British people, in this case the genotype is not the target, but allegiances and loyalties.

This happens in reverse too, and it's not necessarily the Unitists who always instigate it. The Felines are a good example: Hundreds of years ago, Felines lived in individual clans with no real taxonomic or even species governments, and since they identified more with their clan than their species or taxon, the cultures of individual clans started to drift from each other. Some turned Unitist, most doubled down on Trophism, while others remained neutral. The formation of the Feline Empire which united all Felines into a single taxon was instigated by the Trophists, who basically conquered all the clans and enforced Trophism across the entire taxon to prevent the other clans from turning Unitist. Back then, Trophists massively outnumbered Unitists, so the purge happened in the other direction, and any Feline suspected of being Unitist were killed. They would stage hunting tests where if you failed to catch prey in front of the Trophist soldiers in the allotted time, they assumed you were out of practice because you weren't hunting on a daily basis because you were secretly Unitist, and killed you.

Even after these hunting tests were outlawed, the Feline Empire's tyranny and meddling in the lives of Felines was a major driver in turning nearly the entire Feline population secretly Unitist. When they tried to suppress a newly developed dietary enzyme supplement that worked seamlessly on obligate carnivores (obligate carnivores had ways of not eating meat before then, but dietary enzymes made it foolproof and basically impossible to get malnourished on a plant based diet), it was the last straw and caused the Feline Unitist revolution. The Feline empire then drafted every hardline Trophist they could find to fight against the Unitists, but the tables had turned and they found themselves both outnumbered and technologically inferior (again, because the Empire's insistence on the Feline's primal ways being the right way had alienated all the scientists and engineers), the result was pretty much inevitable.

This highlights a major difference in the way Unitists and Trophists reason about killings in revolutionary war. The Unitists would have liked nothing more than to peacefully turn everyone Unitist, but they were almost always fighting against violent and bloodthirsty Trophist armies that were intent on killing every single Unitist they could find, regardless of how peaceful or violent the Unitist individual was, to "preserve the natural order." Once such a revolution had been initiated (and it was rarely a particularly organized uprising, just an overwhelming majority population of Unitists and Unitist sympathizers whose hatred of Trophist system they're forced to live under finally bubbled over), it can pretty much only end until one side is dead, so while the Unitists aren't looking to kill all the Trophists, all the ones that won the war and survived long enough to actually build Unitist societies are inevitably the ones that did their fair share of killing, so you end up with this situation where Unitist societies are inherently pacifist, but getting to that stage involved a lot of bloodshed.

Finally, there are neutral territories and taxa which don't care if a carnivore hunts or doesn't hunt. Even in those societies though, the Trophist population tend to oppress the Unitist population and enforce a de facto Trophism.

(Also, I want to mention that this is not intended to be anything resembling real life socialist theory, and a proper socialist/materialist analysis on my worldbuilding would probably find a lot of contradictions. I'm just trying to tell an interesting story and while it has socialist influences, I'm not specifically trying to write "real" socialism with animals as a top priority. Most of this is developed mainly to let me write action and drama, and a lot of it was developed before I had really understood socialist theory. Just want to make that clear.)

So with this sort of technological advancement, have species which evolved for specific environments (a fennec in a desert) been able to adapt more to pretty much any climate, similar to how we can?

Absolutely! In fact, the Unitist taxonomic governments have Freedom of Migration between their territories, which means members of those taxa have the right to live anywhere else that's governed by a Unitist taxon, and other taxonomic governments are expected to accommodate migrant species with suitable habitat and infrastructure. Again, the end goal is to not have separate taxonomic governments and territories at all so this is seen as a head start to that.

A member of a Unitist taxon can generally live in any climate because their home will be climate controlled to their ideal parameters through heating and air conditioning. This also means that species who have yearly migrations can just choose to not migrate anymore, and even if they do, it will be for cultural reasons as opposed to not being able to live somewhere permanently.

Is the world more of a “humans go extinct and animals pick up the pieces” thing, or more of a world where animals have their own sort of cities, nations and civilizations?

Definitely the latter, at least in Unitist territories. Unitists have highly developed systems for managing and planning their ecosystems to best serve the animals that live there. Trophists tend to do not nearly as well in this regard, despite one of their main "arguments" against Unitism is that the ecosystem "needs" the food web.