This is my area of expertise! I've been studying horse breeds, history, and color genetics (informally, on my own time) for about 40 years.
Curly genes are found in several horse populations, and the weird thing is they're not all the same recessive gene! Someone got a curly mustang at a BLM adoption quite a few years back, and they bred it to a curly Missouri Fox Trotter stallion. The result was a straight-coated horse, and when they repeated the breeding it was always straight-coated, even with one parent that had previously ALWAYS produced curly babies. At least one gene found in MFT's is dominant, while the others have all turned out to be recessive.
I'd imagine without grooming they probably do smell horrid--most of my experience with horses is book-learnin', but I've been around some pretty gross mudmonster horses in my time and they do stink. They groom just like any other horse, though--brush the mud off and run a hose over them (provided they're trained for that) and they're clean as fast as any other horse.
One of the cool things about curly horses is that they produce less of the allergens that keep some people away from horses. They're not TOTALLY allergen-free, but those with mild allergies can often ride and groom a curly without ill effect.
Adult curlies (the one shown is very young) are less curly than foals and yearlings, and they're curlier in the winter than the summer. I've followed some of the adventures of the exported stallion *DCC Vegas, who is a pinto curly Missouri Fox Trotter.
http://jakcurly.com/page-e-vegas-english.html Looks like he's French now :) As the pics show, he is quite smooth in his summer coat. As a foal he was as curly as the young'un at the top of this post, and he curls up heavily in his winter coat. The background on his page is a closeup of his winter coat, while most of the photos with a rider were taken in the summer. You can still see curl in the mane and tail of a summer-slick horse, even if its summer coat is totally straight.
Another well-known curly is the Fox Trotter stallion ZGC Diego, a blue roan who carries the gene that dilutes chestnut to palomino and bay to buckskin.
The Bashkir Curly (or American Bashkir Curly) is, confusingly, NOT related to the actual Bashkir breed, which is a rare western Asian gaited horse (means it has gaits other than the standard walk-trot-canter-run). ABC's are often not gaited, but they do come in a rainbow of colors and are just as hypoallergenic as the MFT curlies even though the genes causing the curl are different.
And each curly can be pretty unique; some have just a "waved" winter coat while others go full poodle. The constitution, stamina, and reasoning ability cited in the Wiki info comes, in many bloodlines, from mustang ancestry. Many of the family lines are now far removed from mustangs, some even going on to be dressage and reining horses (both of which require certain conformation and mental characteristics).
