I will try to be as brief as possible!

Insect farms could be made. Similar to these in China and south east Asia.

Same goes goes for dogs and cats, similar to the food in Vietnam and the two Koreas. Although feeding for the dogs and felines will be required. Therefore, these two would be a luxury food.

Genetically engineered plants will be a must if you want to re-establish settlements then civilization, similar to what the nomads did.

Plants (and fungi which you missed) are the most important part of the food chain as far as multi-cellular macro-scale organisms are concerned. You cannot claim such widespread destruction of the food chain if plants are still there. Many animals would die but with plants and insects around, the basis of the food chain itself would still be there. Omnivores, herbivores, and insectivores would be fine.

The food chain would be pruned, but not destroyed. It's just different than what it was before. It therefore wouldn't need to recover and should just regain its diversity in time as vacant ecological niches are re-occupied.

Rats reproduce extremely quickly and can handle a wide range of environments. This allows to expand both geographically, and into a wide range of niches. Their fast reproduction allows them to evolve quickly into creatures similar to squirrels, mice, and other small mammals. Eventually some evolve into creatures similar to raccoons, possums, otters, and even primates.

Pigeons reproduce quickly, and without aerial predators, they are able to expand as well. Pigeons eat grass seeds supplemented by invertebrates. With less competition, they are able to diversify somewhat into more roles. Some evolve to eat fruit, others evolve to eat small rodents, and eventually into eating small birds. Eventually their descendants fill most of the roles currently filled by birds.

Additionally, pigeons are able to access islands that rats cannot. On these islands pigeons diversify to a greater extant. Large flightless pigeons, similar to dodo birds, are soon the main herbivores on remote islands. With time, they may diverge further, perhaps into something like Emus.

House cats protect human farms against rats, which means that they are given many opportunities to go feral. As they become more dependent on eating rodents for their food, their reliance on humans diminishes. Within a few decades bobcat-sized cats have spread across most continents. In time, some evolve into larger cats, similar to leopards or mountain lions.

Dog are more domesticated than cats, so they tend to stay closer to humans at first. Feral dogs hang around settlements, eating trash and the occasional rodent. In a few centuries dingos evolve from the feral dogs, and in the distant future dogs evolve into wolves.

Feral pigs become monsters, much as they are currently doing in North America. Unchecked, they spread and reshape the land.

The other domestic animals find their niches: mountain goats, mustangs, and so on. Chickens are a bit of a wild card. They could do well, but the rats and pigeons will probably get to most niches before they can. I see them spreading fairly widely but not dominant anywhere, into niches such as that held by the wild turkey now.

Trihorns, well, I don't know much about them. Reptiles tend to do well in warm areas, so maybe some evolve into something like crocodiles. Because of their urine chemistry, reptiles have an advantage over mammals in desert areas, so expect to see a bunch of lizard-like animals in deserts. Limbless and semi-limbless reptiles have evolved several times, so something like snakes will appear sooner or later. The crocodile-like trihorns make gradually become more and more adept at water life, turning into sea creatures in places with warm water.

Let's approach this from a r/K Selection standpoint, mostly because there's a lot more detail that could be desired and a lot of gestation, etc. numbers to crunch if we did it the full predator-prey population cycle way.

• Rat, Insects, Arachnids, Vegetation, & Micro-fauna for r-selected species.
• Humans, Canines, House Cats, Livestock, Pigeons for K-selected species.
• Trihorner (herbivore, omnivore, carnivore? r/K? I'm using it as our joker)

So K-selected species will tend to balance themselves as long as r-selected keeps their end of the deal and remains plentiful. Although it is certainly more stable if it forms a predation loop and/or predator-prey ratios are balanced. Humans can also certainly help dampen extinction-potential population swings by selective hunting or just fully integrate as part of the cycle (your interpretation), as was done in Yellowstone.

As for the r-selected:

• I'm unaware of any place with vegetation that lacks insects. Even a highly specialized insect predator would have trouble wiping out a strong species if it was K-selected. If your insects are diverse then this is even less likely.
• I severely doubt any place with insects would have arachnids dying out.
• Vegetation and micro-fauna can have problems but they're generally fine on their own.
• Rats... as pointed out elsewhere they're a potential problem. They're not generally guaranteed to self-balance since they're not K-selected. And they're able to dine on most, if not all, the other r-selected species. Luckily, if rats multiply by 4-7 every 3 weeks, then that's about 1 additional rat every 3 days. Given that there is 3 potential predators then any population explosion can be checked by them. In the event you got a mega-colony from being isolated in a predator-free resource-rich area, you'd still need an incursion large enough to be triple the local combined population of predators to be a major ecosystem threat (While obviously a different set of ecosystems, in a modern human city you might have barely any predators but plenty of resources. So rats are only resource bound. Our non-urban areas are more balanced related to rats but also supply resources to the urban areas. Cutting off the resources supplied by those more balanced areas you can get things like cannibalism). All of that said though the cat is probably your K-selected guardian of choice for this part of the ecosystem. You probably can't get rid of them, but if you're there to correct an ecosystem imbalance it's not a fight you won't win.

Next is the known concern of over-grazing from livestock killing themselves off. Which could wipe canines and humans out. But given that they are livestock humans managing them can probably prevent that.

Which only leaves the Trihorner to fit into the web. They could very well destroy it as an r-selected so let's make them K-selected if that's alright. If they are:

• Herbivores our livestock have a problem with too much competition. We'll have to hunt them.
• Insectivores (arachnids included) we're probably good.
• Carnivores it'll depends on what they eat. Canines or pigeons makes things more stable. Rats relieves humans of some of the burden. Livestock is probably perfectly okay since we'll try to protect them but will fail from time-to-time. Cats would maybe be pretty bad. It depends on whether canines preferentially hunt cats over rats. Eating humans is not great but probably moderately okay. It replaces the Apex Predator with something with less intelligence to manage the population swings. But it does knock out an omnivore so as long as livestock isn't terribly self-sustainable in the wild the cycle should be okay.
• Omnivores would just be bad, especially if they were an Apex Predator. If their size allowed them to hunt canines and rats preferentially, then no problem. The biggest problem is that they have no extinction clause in this herbivore-dominated ecosystem. Anytime you might have a typical K-selected balancing, predator-prey ratios won't allow it. Anytime the prey is almost wiped out they won't dial back from hunger and reproduction loss, because they can still eat plants and insects. The only check they would have is a Predator Explosion due to the abundance of themselves as Prey. But they would still strain every tier of Prey underneath them.
• Apex Predator (see omnivore and carnivore)
• "Other"-vore (fructivore, etc.), specialization makes this less likely to survive as it hinges on the survival of something else. The more specialized the less likely to survive (a "problem" that is, in general it'll be more efficient at what it does and flourish). However it also is highly unlikely to mess with the rest of the ecosystem, probably at most it would knock itself out with its Prey.

So will this food chain survive?

If the trihorner isn't there, then probably. With it actually present though it will need to meet some conditions. Maybe strict preferences in diet, maybe specialization in hunting a particular insect, maybe it eats all meat except human meat. That said, humans, as always, will have an amazingly large potential to upkeep or royally screw over the ecosystem.

If I remember correctly, the common ancestor of mammals during the times of dinosaurs extinction was something small similar to a rodent.

It looks like you are in a similar situation: rats, lacking predators, will proliferate and occupy all available niches, then, to mitigate food competition, they will start specializing on a certain diet, which in the long run will separate them in new species.

It won't happen in a few centuries, but that's how it might go.

Pigs and chicken, among the livestock, are versatile enough to be able to start their own species trees, too, if they manage to escape human control.

Predators are sorely lacking

Without predators keeping them in check, anything that rats can eat, they will eat into extinction. Then the rats will starve. Canines - i.e. wolves - are the big exception here. Basically any mostly wild area will be full of wolves, living off of mostly rats, or no animals at all.

People, who regard rats as pests, will keep the population in check but not be able to wipe them out (we never have before, have we?). Therefore human settlements will have a constant rat population. This means the area surrounding any settlement is either full of wolves, or a barren wasteland.

You're sure there's no birds?

Lot of kinds of birds live in human cities.

Many plants go extinct from the lack of codependent animals

Many many plants are adapted to take advantage of the existence of animals to spread. Burr seeds get caught on animal fur, then eventually fall off, the seeds spilling out potentially a great distance away from the source. Berries and fruit are designed (insofar as evolution designs anything) to be eaten, and then the seed survives its typical eater's digestive tract, and gets deposited elsewhere.