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.