Let's remember 2 things really quickly: as far as we know, ever single vertebrate in our world can be essentially traced back to a single ancestor and we had dozens of extinction events throughout the history of our planet. Right after these extinction events, we usually see a large amount of niches available due to the death of the previous occupants and a relatively small number of surviving species, which in cases can result in speciation booms which allow for new species to appear and fill in the gaps. However [that's not quite the entire story](https://evolution.berkeley.edu/evolibrary/news/120901_afterextinction), why? Because some species are capable of faster speciation than others, for example, an animal that has more offspring at a faster rate can undergo changes and select beneficial mutations much faster than a species which has 1 or 2 children every 2 years.

So let's have a simplified analysis about the remaining species regarding predation and temporarily assuming plant life is not a problem (because I'm no biologist and not all factors are known about your world, plus you seem to be intending to focus on the terrestrial environment):

- The surviving arthropods seem to have no remaining predators other than one another, rats, chickens and an eventual pigeon unless humans too begin using them as a food source. 

- Rats, unless the dogs and humans start using them as food, only have to worry about chickens and pigs (a pig can and will eat a rat if it's hungry or if it stole the pig's food, a chicken can and will eat a rat period, both these farm animals are omnivores after all), however despite their ability to reproduce fast, it's hard to say if they'd undergo a speciation boom, since lifespan is also an important factor, and a rat will hardly live more than 4 years even in captivity. Rats may eventually eat one another in times of hunger as well. 

- Your cows, horses, llamas and donkeys superficially have little to worry about other than a large pack of hungry wolves and the now relatively uncontrolled (depending on how the remaining chickens and pigs are doing) rat population (and humans of course, though humans might actually help them survive in the long run by doing what we've done for centuries, save them from other predators so we may be the ones to eat them).

- Sheep might be at a greater risk of being attacked by wild dogs, but seem to be on a relatively similar situation as the other large ruminating animals, assuming humans keep using them as livestock.

- Goats might be similar to sheep, except they might actually be desired since they've [shown to help controlling rat population in some cases](https://www.pestcontrolnews.com/pcn-interview/).

- Regarding igs and chickens, apparently a mostly similar story, except they have the potential to act as predators (to rats and, if the pigs are starving, whatever they can catch).

- Humans seem to have mostly the environment, the rats, an eventual pack of wild dogs and the trihorners to worry about and seem to be better off making their best to keep the livestock strategy going and plant whatever edible grains are left. Keeping dogs close by also seems like a good choice.

- The dogs will likely be split into 2 groups:those who keep living with humans and those who go wild, trying to fill the niche of their wolf cousins. They'll likely predate mostly on sheep, some pigs and goats, as well as some chickens and an eventual rat, if they're hungry and it's easy enough to catch. I don't see too many attacks on larger livestock unless the dogs belong to breeds more optimized for hunting and are in large enough packs, and their causalities will likely be predominantly due to failed attacks, human/domestic dogs intervention and potential predation by a trihorner. 

- The trihorners: you didn't say much about them other than that they're genetically engineered, so I'll work with some assumptions and what was discussed in some of your previous questions: assuming a warm enough climate, your trihorners, assuming they're slightly larger than a komodo dragon, will likely compete for the position of Apex predator with wild dogs and humans, depending on how much these reptiles were engineered to resemble a living weapon. Assuming a deathclaw scenario (created to look like a salt water crocodile and a komodo dragon, both on steroids, somehow had a mutant baby which was feed nothing but protein shakes and more steroids) your Trihorn will easily occupy the niche of solitary yet powerful reptile and occupy the top of the food chain. Dogs might be a problem for the young and the elder, but a healthy adult Trihorn, similar to moose, will likely have no natural predators and will only need to watch out for humans trying to hunt it (or not, depending how much the "on steroids" factor is present in this creature).

So overall, considering the animals only, the environment as is seems to be already able work out relatively fine (although the number of available potentialy predatorial species seem smaller than ideal), assuming plant life is doing well enough and the remaining animals are already present in necessary concentration. If not, we will have problems regarding overpopulation of some species followed by rapid decline in said species, though unless something like the rats runs completely beyond control, I think the environment can regulate itself enough with what's available. What I mean is that, while there are many niches which were left unoccupied, there are still some animals helping in population control,and seeing how mammals all seem to have come from a single ancestor, all vertebrates seem to have come from a single species of fish and we've had a fair amount of extinction events already, we can assume that the most likely scenario is for some of the species still present to slowly undergo specialization to fill in niches left by the extinct land vertebrates. Which species will undergo such speciation is hard to tell, but my bet lies on mostly the dogs, pigs and chickens, as they can all reproduce relatively fast and will all usually live more than half a decade (unlike rats, which reproduce incredibly quickly but don't live as long, though they too might give origin to new species).