how a town slime arrives
Town slimes are not rescued and they do not simply walk into an empty house. Almost all of them are made by converting an ordinary slime, the same way the Tax Collector is made rather than found. Once that conversion happens it is done for good: the new slime heads off and moves into any vacant house you have available. If something kills it later, it returns on its own after a while as long as a house is still free, and you never have to repeat the transformation.
housing them cheaply
A town slime will share a house with a regular NPC, so it costs nothing beyond a room you already built. It will not share with another town pet, and a cramped shared room can still turn it away, because it needs a free spot of its own to stand in. All eight can live in the same world at once, unlike the cat, dog and bunny, where the breed you pick is the one you are stuck with.
Housed pets count toward the two nearby NPCs a pylon needs, so tucking a slime into an existing house is a quick way to prop up a thin outpost town.
getting the other seven
Their conversions are scattered across the whole game, which is most of the appeal. Squire Slime only accepts a specific list of ordinary slimes, present slimes and Pinky among them, and a slime spawned from a statue is a legal target, so you can manufacture the input instead of hunting for it. Surly Slime is fished up during a Blood Moon and arrives on the line in red like a hostile catch, though it will not attack you; a Sonar Potion confirms it before you reel in and a Chum Caster improves the odds. Diva Slime is the only Hardmode exclusive, made by shimmering a Sparkle Slime Balloon, and it is already living in your world from the start in a Zenith seed.
where the design came from
The whole group is built on the Slime Prince and Slime Princess, the Master Mode pets that King Slime and Queen Slime leave behind. Most of them reuse that body shape with different colors, hats and expressions, and the Cool Slime wears the Prince's face exactly. Diva and Mystic both glow in the dark. There is also an unused line in the game files about a license teleporting off to a slime delivery service, which suggests these were originally meant to be bought the way the other town pets are.
names
Every town slime name begins with S, which the developers have called mostly coincidental. The Nerdy Slime draws from the obvious well, so you may end up housing Spock, Sputnik, Shibboleet or Sine. The function that creates it in the game's code is named ViolentlySpawnNerdySlime, which is considerably more dramatic than anything the slime does once it moves in.