First thing to note about polar and high-altitude environments is that, from a physiological standpoint, they are very difficult places to live. Creatures which live there are going to be one of two things: Evaders or Endurers.
Evaders are generally small animals that make use of behavioural strategies to avoid the harsh conditions (which, in both habitats, includes below-freezing temperatures, low humidity, lack of food, and - at high altitudes, or when there's snow on the ground - dangerous levels of UV radiation. They're going to do this by exploiting microclimates within their habitat - mostly burrows (hard to dig in frozen soils, but tunnels under the snow are both significantly warmer and significantly more humid than the environment above) or rock crevices (especially in alpine habitats, where erosion creates many ideal hiding places). Evaders are going to time their behaviours to coincide with periods of lower environmental stress - e.g. hibernate during the winter and move around during the summer, or only leave their burrows at daytime when they can warm up in the morning sunshine. Another category of evaders are migratory species which live and feed in the polar regions in the summer, and run away to more temperate climes during the winter. These animals can be larger, since they don't need to exploit microclimates when conditions are bad - they can just leave. Being big is also good for travelling since you can store more energy as fat reserves and cover more ground. Most animals in extreme environments are going to be evaders, just given the relative complexity of adaptation involved.
Endurers, on the other hand, don't migrate and are too big to hide in burrows or rock crevices, so they just tank the environmental damage incurred by the cold temperatures, low humidity, and lack of food. These tend to be large animals that can't effectively hide in burrows or crevices, and which are forced to face the elements head-on. Think muskox, etc. These animals must have specific adaptations to weather the cold - namely, endothermy. Looking at the distribution of life on earth today, the only animals which have any kind of significant presence in extremely cold environments are endotherms - lizards, frogs, salamanders, snakes, etc. all reach limits of their ranges much further south than mammals and birds. Polar endotherms must have additional ways of keeping warm, including thick insulation (blubber or fur), round bodies with few extremities (which are vulnerable to frostbite), and a complex system to warm incoming air so it doesn't freeze the lungs.
In the carboniferous, true endothermy is going to be rare (if it exists at all) and thus very few, if any, animals are going to be endurers. You might get some giant amphibians in high-latitude temperate swamps with similar adaptations to large pelagic fish like Tuna, but that's about it unless you find a clade that can swing both endothermy and insulation. The vast majority of polar animals are going to be evaders. Maybe some primitive synapsids or flying insects will be able to make the annual migration; more likely, the small reptiles, amphibians, and arthropods are going to wait out the winter in burrows and tunnels. Look at modern temperate ectotherms as examples: wood frogs can freeze solid in winter, garter and rattlesnakes find deep holes to hide in until it warms up again in the summer. In the absolute worst of conditions, on the ice sheets themselves, you might find some cold-hardy arthropods (like springtails) grazing on bacterial or algal films (maybe overwintering as cryptobiotic eggs or cysts), but for the most part there's gonna be nothing.