You may wish to ask this question to r/AskAnthropology as well as what you are asking isn't strictly a "historian" question. To put it simply, history involves the written word, and the domestication of the horse is something occurred far before the invention of writing. That being said, having read into this topic, I can give you an answer too.

Firstly, on the claim of Indo-Europeans taming horses in Central Asia. All the evidence that we have available today suggest that this is valid. Based on archaeological evidence (in particular bones), far more horses were present on the Eurasian steppes (southern Ukraine to Kazakhstan) than elsewhere in Eurasia. This isn't to say there weren't horses elsewhere (although they did die out in North America), they just weren't as numerous. Great Britain, an island by the time of domestication, had its own horse breeds native to it. Likewise, there were also equine populations in France, Spain, Germany, and Turkey to name a few. These populations were mostly isolated pockets, seemingly limited by the boundaries of the meadows and pastures they inhabited. Only in the Steppes was the population far reaching and numerous.

Much of our data for determining this comes from human garbage. Ever since the dawn of time, human beings have thrown out what they didn't have an immediate use for. The frequency of horse bones in these garbage heaps give us insight into the animal populations that made up the diet of ancient humans. Since horse bones are far more numerous in the garbage of the steppes, it is highly suggestive of a large equine population and very clear evidence of human-horse interactions (even if it was human-eat-horse interaction).

Now you may ask, what do large populations of animals have to do with taming? David Anthony in The Horse, Wheel and Language, suggested that "domestication is the culmination of a long prior relationship." He argued that people would not invest time and energy into animals they didn't know much about. Small, isolated pockets of horses would not have given man enough husbandry experience to decide to try to tame them. On the other hand, the seemingly inexhaustible numbers on the steppes would have made steppe cultures very familiar with their equine neighbors.

The understanding of equine behavior would have led into the decision to keep, feed, and raise tame horses. Riding horses would not have been on the minds of early man when domesticating the beasts. Horses had long been hunted for food prior to domestication, and they continue to be raised for food even to the present day. Furthermore, horses tend to fare better than sheep and cattle at weathering winter conditions. Where cows will use their noses to push away ice and snow to reveal grass, rubbing said noses raw and bloodied; horses will use their hooves and not suffer adverse effects. This would have made horses fairly attractive as a winter food source and worth the gamble to tame.

TL DR: Yes, the Eurasian steppes is the accepted location for the domestication of the horse.

As for your second question of horse burial, it is slightly problematic in that horse burials are not unique to the Hittites or the Egyptians. I can list a hodgepodge of cultures across thousands of years of history ranging from Dark Age Greece, the Romans, the Scythians, the Chinese, to the native cultures of North America (after the reintroduction of the horse) that have all practiced it in some form or another. It is too broad for me to answer, and quite honestly deserves an entire book on the subject. Do you have a specific culture in mind that you wanted to know more about horse burials?