what species could they have evolved from, and what evolutionary pressures would lead to them?
No species that currently exists. The necessary combinations of features just don't exist in any animal; you'd have to go back to the Permian period to get something that could branch off into a whole different biological class, in between mammals and birds.
If you are willing to relax the "birdlike wings" requirement, however, I could see it evolving from some kind of large prehistoric bat. Say, bats subject to Island gigantism. As they get larger, and in the absence of a lot of predators, the need for flight would reduce, and the may develop a body plan similar to a pterosaur, using the wings to walk and prop up the torso, so they can occupy the browsing niche of a giraffe, but with ridiculously long front limbs rather than long necks. Becoming larger and shifting towards a browsing diet would then lead to developing a more bovine or equine face and tooth set.
Now, have a land-bridge open up or something that introduces these quadrupedal mega-bats, who presumably can still fly but usually don't bother, to a bunch of new predators from whom they need to be able to escape. There are many ways to deal with that, and I expect you'd see a bunch of related species around to give clues to this origin for your wyvern-pegasus, but one option would be to just get faster--shrink down a bit, and lengthen the hind legs, so they can gallop more efficiently, because even if you can fly, running away from a predator is cheaper when you are that big. Which is why, e.g., ostriches and rheas go in for the running thing. Again, once you start going down this path, expect a split--some groups might get small enough that just using their now-elongated back legs to jump into the air and fly away is in fact the ideal option for escaping predators on the ground, but bigger ones may discover the benefits of bipedal running, using their wings for balance and steering.
Somewhere along the way, these mega-bats may have developed hoofs on their hind feet, just like actual giraffes and deer and goats and cows and so forth. Most likely, you're going to get a cleft hoof, as horses are the only creature on earth that has developed a single-toe hoof, and that seems unlikely to repeat. However, even if they somehow avoided converging on the large-mammalian-herbivore solution of a evolving a hoof prior to this point, getting good at running fast is likely to result in evolving a single large central toe at this point (which you can also see on ostriches, so that bit is not limited to quadrupedal runners), at which point a cleft hoof might turn into an effective monohoof with an extra small toe, or you just finally evolve a hoof at this point.
That gets you an animal that is normally quadrupedal (because it does not have birdlike wings, and can in fact support itself on its palms) and grazes on grass, shrubs, and low tree branches, but which lifts its wings to become bipedal when it needs to run at high speed, and as a last resort can, with good running start, lift off into proper flight--but not if it's carrying anything (or anyone). Perhaps they even regularly fly to cover long distances efficiently, through soaring--e.g., for seasonal migration.
Meanwhile, it would have close cousins that are smaller and completely incapable of being ridden, but which regularly fly, and other cousins which have stuck with the giraffe niche, stayed large, and adapted to the presence of predators by getting tough and nasty rather than learning to run more efficiently.
That takes care of having fur, being slightly smaller than a horse (but considerably lighter, of course), being quadrupeds, being herbivores, and being capable of flight as long as they aren't being ridden (because they are right at the upper end of the weight range where they can still fly at all). Being monogamous is kind of "whatever"--there's no reason they shouldn't be.
Having very downy fur and having a precise torso and head shape between a Przewalski's and an Icelandic horse can easily be explained by someone domesticating them and deciding that those are aesthetic features they like and want to breed for.
Being mostly solitary is a little trickier. Large grazing animals tend to come in herds, after all... unless they are male. So if you're OK with half of them being solitary, that's easy--just like rhinos or elephants, males get kicked out of the herd when they approach maturity, and live mostly solitary lives until it is time to court a mate. Paired with the monogamy feature, you could have isolated males seeking out small herds of females (consisting of just a mother, father, and children, rather than huge multigenerational herds) in order to try to impress one and draw her off to found a new family/herd (which is what keep herds monogenerational and small).