Fair Warning: Spoilers galore below.At first there were brown ur-quan and Gg and ect and he precoursers, everything was fine,
The Precursors inhabited our sector of space approximately 250,000 - 300,000 years ago, long predating any other known race. They disappeared about 200,000 years ago, for reasons that no one has yet discovered.
The brown Ur-Quan reached the stars and joined the Sentient Milieu about 25,000 years ago, which is still long after the Precursors disappeared.
The Gg were an enigmatic planetbound race whose only extra-planetary contact was the Burvixese. Both of these races were annihilated by the Kohr-Ah shortly before the beginning of Star Control II. Neither one ever had contact with the brown Ur-Quan or any other race from the Milieu (except for the Melnorme, of course, and they don't really advertise their ancestry.)
then the Dynarri or those Talking pet things came and enslaved he ur-Quan, there were probably groups split into different classes based on what the pets wanted to get done.
If by "came" you mean "the Ur-Quan found them on their homeworld during a routine scouting mission" and by "split into classes" you mean "they genetically engineered the Ur-Quan into two different subspecies designed for specific tasks" then yes, this is essentially correct.
Somehow, I don't remember how, the Ur-Quan broke the mind enslavement,
A green ur-Quan scientist byt he name of "Kzer-Za" discovered that the Dnyarri would release a slave shortly before its death, otherwise they would die as well. This lead ultimately to a revolution wherein the Ur-Quan would voluntarily mutilate themselves in a variety of ways, gaining momentary freedom in the process, and was the beginning of the Slave Revolt.
maybe with some kind of taalo device
The Excruciator - a device designed to inflict agonizing pain on it's wearer and a key element in the success of the revolts - was a device of Ur-Quan design. The greens later adopted the name Kzer-Za for their subspecies in honour of the scientist who gave them their freedom.
The Taalo Shield was something designed by the Taalo race to protect their allies in the Milieu. The Taalo had a natural immunity to the Dnyarri's mind control and were endeavouring to develop a device that could extend that protection to other races. Unfortunately, the Dnyarri destroyed them (via the Ur-Quan) before they had a chance to share what they had built.
Paul and Fred have since made comments in interviews with fans that suggest the Taalo are still alive in some vague and nebulous form, however they don't appear to be active in our region of space during the game, so this is presumably part of a plot thread they had planned for a sequel they never got the chance to make.
and with Asked he Umgah to engineer them, which explains not only why there was an extra one at their homeworld but also why they were allies.
The Umgah are a species native to our sector of space and also to our era of history. They were not around during the time of the Dnyarri Empire. They work primarily with genetic material, where the Excruciator was (by all accounts) a technological device.
There was no Excruciator or any other ancient artifact on the Umgah homeworld. The Taalo Shield was found on the Taalo homeworld of Delta Vulpeculae II-C, which is not even within Umgah space during the course of the game. The device the Umgah had at their homeworld was a HyperWave Broadcaster, a very common device in modern times used to allow real-time communication across interstellar distances. Thiers was a particularly powerful model, but otherwise it not really noteworthy.
Finally, the Umgah and the Ur-Quan are not "allies." The Umgah chose to become Battle Thralls rather than slave-shielded Fallow Slaves when the Ur-Quan conquered them, but this does not mean they are in any way equals to the Ur-Quan. Nor is there any evidence that the Ur-Quan have ever availed themselves of the Umgah's genetic engineering talents.
But after that, they were all paranoid or something and decided to try and enslave every threatening race so that nothing like hat would ever happen again, only the dichotomy within the species lead to the destruction of the Gg and a couple other races
All of the races from the Sentient Milieu who were exterminated by the Ur-Quan were destroyed while they were still enslaved to the Dnyarri. Of the three non-Ur-Quan races who survived the Dnyarri Empire, the Faz were slave-shielded by the Kzer-Za, the Yuptar were annihilated by the Kohr-Ah and the Mael-Num hastily escaped while the two Ur-Quan forces duked it out in orbit above their homeworld. The Ur-Quan were indeed paranoid about the idea of being enslaved again after having thrown off the Dnyarri enslavement.
The Gg, as discussed before, were killed by the Kohr-Ah much more recently and long after the two Ur-Quan subspecies had begun their Doctrinal War.
and this is when the precoursers decided to get the hell out of there,
Nope, as discussed above, that happened long before the Ur-Quan even existed as a star faring race, and no one knows the true reason for their disappearance.
then over time other sentient races became more visible and grew larger to what is now seen in the game, only the dichotomy of the Ur-Quan still remained, though most other races were unaware as the Khor Ah's boundaries were more to the upper right and the Suppox and Utwig did not have as large territories.
The sentient races that currently inhabit our sector of space did indeed grow larger over time, however, they only managed to do this because the Ur-Quan were gone, off circling around the galaxy for the last 25,000 years or so. The Kzer-Za only returned here a couple hundred years ago and began the process of enslaving all the races who had sprung up in their absence. The Kohr-Ah arrived somewhat later (closer to the beginning of the SC2 game events), annihilating a few species on the way in, before they found the Kzer-Za and began the Doctrinal War.
Remember, the influence spheres for each race seen in game only represent the area of space each race
currently occupies. They move around and grow/shrink as a race's influence changes, so just because the two Ur-Quan spheres are where they are in game doesn't mean they've been there for the last 25,000 years.