First off, the basics:radiation isn't like the mutagen thing we see in superhero stories and tmnt. Being exposed to large doses will most likely cause mutations, but these are absolutely random. Radiation won't twist you into an alpha predator that can shake off tank rounds. Just look at the theory on death claws from the fallout universe: many assume they were modified before the apocalyptic nuking, and what the radiation did was kill off most predators, opening up a niche in which it could thrive. Just look at how the animals in chernobyl have been doing in the last 30 years, with most radiation-induced mutations happening mostly at the beginning. As I see, the point of the radiation here, rather than causing animals to evolve fast, is to kill off various species so that the remaining ones have many available niches to occupy, thus evolving rapidly (this has actually happened more than once in history, usually after large extinction event's).
Now, let's analyze the creatures to see if we can make them work. I'll assume here a long enough amount of time has passed, since it takes time for evolution to do its thing:
Rad-rats: sound reasonable enough. Maybe not the size of a capybara, but given that polar species do have a tendency to become larger in order to withstand the cold, it sounds reasonable enough that larger rate which were more active and grew bigger were selected over the smaller and less active ones, which froze to death. Being vicious means they're either very territorial, have adopted a carnivorous lifestyle or both, which again isn't too crazy considering the land is now cold, plant matter isn't as available and thylacoleo exists, so we do have records of animals the fed on plant matter adapting into predators. Your rat essentially followed a path similar to the polar bears, and will either need to adapt again if the climate becomes warmer or migrate to colder refions in which it can survive.
Trihorners: these are very unlikely. One thing you'll notice when looking at the animals in the north and south poles is that both reptiles and arthropods are mostly absent. If the world temporarily became as cold as the poles, if not colder, then most likely the majority of, if not all, reptiles has either gone extinct or, like we already observe in a few species, started relying heavily on their own ability to produce heat. For reptiles to survive, they'd need to evolve to become both homeotherms and to develop protection against the extreme cold, which is unlikely. The only way I'd see these creatures working is by ripping off the deathclaw: a group of scientists trapped in a bunker with their giant vivarium of reptiles, free from a my annoying ethics protesters and having plenty of available resources, decided to play God a little bit. The result? A creature created through the association of many animals, from crocodilians, to komodo dragons, to the jackson's chameleon. The result? A predominantly terrestrial, quadrupedal reptilian marvel,having 3 sturdy (although smaller in proportion) horns of the chameleon and the powerful tail of a marine iguana, it has the heard of a crocodilian, being much more active. Its imposing size, as large as a komodo dragon and with venom glands to match, this fully viable new species of creatures is more than ready to go out with its creators and become the first genetically engineered species. They just didn't expect the rad-rats that attacked them soon after they left. Oh well, their lizards still made it out into the now warm enough land, and are doing quite well. Their prey on the other hand, is having problems.
Giant scorpions: nope. Ain't gonna happen. What you're asking did already exist in a way...as an aquatic creature which lived at a time in which oxygen concentrations were considerably higher than modern in days. Your land scorpion's main enemy is something that has plagued both world builders which want to be somewhat realistic as well as their creatures:the square cube law. Even if there was another group of scientists trapped with their vicarium of arthropods and plenty of resources, unless the atmosphere has reversed to its Cambrian composition, this thing won't be able to survive (a giant pangolin, on the other hand, maybe could work out wink wink).
skinless: not in the way you say. You might not know, but the skin is a vital and one of the most important organs in our body, shielding us from pathogens and radiation, helping maintain our temperature, helping us avoid dehydration etc. A creature without skin won't 1-survive a nuclear winter 2-survive the vast amount of bacteria and other pathogens infecting it unless it has one glorious immune system, and those are just the most obvious issue. I present to you an alternative: the nightwalkers, primate-like creatures with tough skin and fur, much like a sea otter's coat and have a dark color both due to their nocturnal lifestyle and to protect from radiation (higher levels of melanine in its cells and fur). This creature relies on large eyes to see though the night, and it's fur help blend in with its surroundings. They're rare because their population is small, as well as due to their pristine camouflage in dark environments. These strong primates. Probably having come from chimpanzees, and potentially being another job of scientists and geneticists playing God, as they seem to have come from chimpanzees due to their intelligence (what's with scientists in these post apocalyptic scenarios?), these animals are violent and stealthy, fitting well the vacant niches of nocturnal predator and scary tale for children.
Summing up:can they exist? The rats? Yup, they're the easiest one to happen and aren't too hard to believe in. The reptiles? Not naturally, unless they undergo several mutations to survive the period of nuclear winter. My best bet lies in geniuses of biology and genetics trapped in a giant bunker and bored to death seeing what they could do before thinking about heat they should do. The scorpions? I don't think even genetic modifications can make that work. Skinless? Same as scorpions, those zombie like creatures won't do without some metro style paranormal help. A nocturnal predator however, could maybe work out.