"it appears to be implying by the photos that horses started out very tiny, and with evolution became larger. Is that an accurate statement?"
Yes. The sequence of creatures attributed to the evolution of the horse start off very small. You start off with a very small, very generalized browsing animal. A foot or so tall and less than fifty pounds. The descendents go in all directions. Some toward horses. Some rhinos. Some tapirs. And some other, now extinct, animals. This generalized animal was probably Tetraclaenodon. From this you get two genera, Hyracotherium which led to the horses, and Homagalax, which led to the rhinos and tapirs. These two genera differ only in some very specific details of the teeth.
"Also, perhaps I'm missing the link, but didn't you say there were ones that showed fossils of what these horses were before they were horses? These skeletons on the site still look really horselike to me! Except for the ones that look like dogs. LOL"
Well... Like I said, only the last genus, Equus, is really considered to be a horse. I was trying to give you the skeletons to show you that many of these are known from nearly complete specimens. But it is difficult to get a feel for what something looked like from the skeleton. Especially since the reference I provided scales them all to fit the same size box. I might could find some artist's representations, but those can be suspect.
But you are on the right track. I am not sure that I think that they look like dogs. It would not be completely unfair to say they look like little horse with toes and claws and a short neck and a short snout and an arched back and some really flexible legs. No worse than saying a bear looks somewhat like an overgrown wolf / raccoon hybrid.
The earliest examples, eohipus, orohippus and epihippus were somewhat similar with changes happening in the teeth largely, but other places too.
With mesohippus and miohippus you start seeing some other changes. Again, the teeth are prominent. It is during this time that they archieve the basic characteristics of the horse teeth. They also are getting a little larger, now around 2 feet tall and the head / snout begins to get a bit bigger. By the end of this period, you have an animal that has gone from a generalized browser to one suited for grazing on grass, which is very hard on the teeth. This is also the beginning of the move towards running ability.
Once you get to this point, changes start happening as the creatures become more suited for grazing on grasslands. The teeth continue to modify for chewing grass accompanied by a lengthening head. For running on grasslands, the body gets bigger, the legs get longer, and the animals start standing / running up on the toes instead of the pads on the feet. Some animals from here are parahippus and merychippus.
At this point you have a lot of different "horses" running around. You would likely even recognize them as horses if you saw them, though they are not yet what were known as true horses.
From here you have an evolution towards the horses, zebras, donkeys, etc. of today. You get the classic horse traits of "rigid spine, long neck, long legs, fused leg bones with no rotation, long nose, flexible muzzle, deep jaw." Now that they are running up on their toes, the claws turn into hooves and you see the loss of the two side toes as they come to use just the center toe. The size gets to the modern size.
"For example, I'd imagine if they came from a bird, there would be a stage where they had wings If they came from sea animals there would be in between fossils of horses with scales or other properties."
Well. they were just browsing, placental mammals.
"What was the dawn horse before it was what is being called a dawn horse?"
Well, I think we stepped through that. The common ancestor of the horses and rhinos was part of a group called perissodactyl, which just means that they are / were ungulates (hooved animals) with an odd number of toes. (The Artiodactyls are even toed ungulates and include sheep, goats, camels, pigs, cows, deer, giraffes, and antelope.) Before this came the condylarths. They date back to the time of the dinosaurs (give or take a little, there are still some questions) and are not known from much more than teeth and jaw bones. The teeth though were beginning to look like the teeth of later animals. There are some possible earlier ancestors, but now you are getting back close to the origin of placental mammals in general. So, for simplicity, just think of a rat. You are close enough for government work.