1. Those chromosome numbers for horse and donkey of 64 and 62 are diploid numbers meaning they received 1/2 from each parent. The hybrid mule receives 32 from its horse parent and 31 from its donkey parent totalling 63.
The completely hypothetical evolution of donkeys into horses may be explained in only one mutation event. Imagine one chromosome of a donkey gamete split in 2. The offspring would have 31+32=63 chromosomes. If this offspring was still fertile (this isn't a mule yet) it may supply 31 or 32 chromosomes. But imagine it passed on 32 to a few children, and some of those children passed on 32 to some of their children. Then some of those distant relatives bred and each passed on 32 to form 64. If this new combination was also fertile and fit (more fit than the 63s and even normal 62 donkeys), it would begin to increase in frequency. The breeding population that it belonged to would over time be dominated by 64s to the point that 62s and 63s would be extremely rare. Over enough time to breed on their own, the 64s depart even further from 62s until when they did come in contact again they may no longer be compatible.
Ring species are an example of this. There is a salamander in California that lives in the mountains. The mountains form a ring there where there is a coastal range and the inland Sierras. The salamanders have a color gradient along this ring where in one direction they are more uniform reddish, and the other they are darker and blochey. The salamanders are reproductive all along this gradient except for one region. In this one region, the extreme red variety is in contact with the extreme dark blotchey variety and they are not reproductive. It seems to be that the salamander was originally isolated to one area (opposite this ) and radiated outward in each direction. As they migrated in each direction, there were different selective pressures for their color, such as need to hide from predators. Over small distances, these changes are small, but the species radiated out around the mountain ranges until finally the 2 distinct radiations came in contact ( many, many generations later ) and were no longer reproductive. The classification as separate species is very sketchy in this instance because of this gradient. But if there was a natural event that severed contact of populations on this ring, two groups would become fully isolated from one another (the existing cohabitation region without reproduction is the 2nd boundary). Over time these two groups may be completely unique even if the severed region was reconnected. In that case, it would be clear that there are now 2 unique species.
2. The "original" creature does not seem to exist in those set boundary terms that you have described. Yes technically there must be an original mutant but this mutant just must continue to live and breed with the existing population. It may be virtually indistinguishable. But if the mutation is lucky or favorable it will increase in frequency and eventually could even be a requirement.
Regarding the separation of sexes, there are numerous "stable solutions" to reproduction. They are just that; solutions to a "problem". That is, how to not go extinct. When it comes down to it, that's all that matters. Once you go extinct, it's game over. But before then, whatever works is fine. So populations found numerous solutions to this problem. There is sexual reproduction (separate organisms or selfing hermaphrodites), and a multitude of other mechanisms such as parthenogenesis, fission, sporogenesis, budding. Some organisms seem to be very flexible with this and the number of chromosomes. Quite a few plants have a lot of chromosomes, probably through some kinky sex stuff in their past. Any way, with evolution it important to try not to fall in traps of believing there must be a reason for something. If you flipped a coin and placed a wager on the outcome, technically there was a reason for the outcome (you flipped the coin with this force starting in this position and measured it at this point) but this reason is completely detached in meaning from the outcome that you experienced. So evolution is best thought of as a loose collection of mechanisms with only one punctuation: extinction.