I've never had to deal with the Korath before. Are they the only ones who get spore ships? I thought they were until i noticed that there is a template for a terran spore ship. I guess you can trade the tech from them?
No you can't, but you can create a custom race that uses terran ship style, and have the super exterminator super ability.
So if you are the Korath and you use a spore ship, do you then need to send a colony ship to the planet? Can someone else claim it before you do? If it becomes a Korath-only planet, is that for the entire remainder of the game?
Nope, you get the planet with an almost non-existant population. It makes it easy to get your planets back from them quickly (or to lose them as the Korath). Their freshly exterminated planets are so vunerable, I've actually seen the Korath in my games switch to normal invasion tactics. Clearly the AI can realize how quickly they can lose their spoils of war.