How long does it take for a foal to become full grown? How long would it be able to be a traveling horse before it can't really do that anymore? Would a child be able to do any training with a foal?
Most people won't start a horse under saddle until it's at least 2 years old. 3 or 4 years old is pretty common as well, but they continue to mature and fill out after that- with some breeds (especially the taller or heavier breeds) taking longer than others. Aside from physical maturity, the younger the horse is, the shorter its attention span is. With young horses, it's better to keep training sessions short and frequent rather than try to do long sessions with more time in between. Before riding, horses can definitely be taught ground manners and even tricks; with a foal, it would probably be mostly how to lead, accepting being touched all over, etc. A horse that's lightly ridden and well cared for could potentially be ridden into its mid-20's or later, but the harder it's used and the less optimal its care, the younger it will be before having to be retired- even as young as 15.
What does "rubbing down" mean? What's it's purpose? What's the purpose of a horse blanket?
After a long, hard ride, horses get pretty sweaty, so it would be normal to hose off a horse (or in the case of your story, perhaps sponge them off or something similar). A blanket can be used to cool down a horse more slowly so they don't catch a chill, or can be left on during cold weather, but this is often more for the human's peace of mind, as a healthy horse allowed to grow its winter coat naturally can handle temperatures down to -40.
If you're only using one horse to travel long distances, how often would you need to take a break or slow to a walk to ensure you don't hurt it?
Wild horses cover a lot of terrain, I've heard upwards of 20-30 miles per day, but they are foraging for food while doing so. A person riding wouldn't want their horse to be meandering around picking at grass. Instead, they might ride for a few hours at walk (3-4 mph) with some trotting (8-10 mph) then stop to water the horses, let them graze, etc. while also taking a break themselves. I would guess a horse that's conditioned to that kind of travel on a frequent basis could probably sustain 25-30 miles a day as long as it's easy terrain.
What's the best kind of horse breed for a flat plain with extreme weather?
I'd avoid naming a specific breed; instead, just describe their characteristics and let the reader assume it's a horse bred for those conditions
I picture them being very mobile, fast-paced warriors, focusing more on archery than anything else, so their riding would probably place a lot more emphasis on being able to steer and give commands with your voice and your legs.
Since you mention archery, you should consider whether or not the people in your book ride with stirrups, as the stability they provide to mounted archers has played an important role in the outcome of many wars throughout history.