Rat

Rats only appear in a Graveyard, and a Graveyard is something you build rather than find. Cluster enough tombstones in one spot and the biome forms around them, the music changes, and the critter list changes with it.

Any bug net will do. Carried loose in your inventory it stays a rat you can release later. Turned into the cage it becomes permanent decoration, and normal crafting will not undo that, so keep a spare loose one if you like rearranging pets. Shimmer is the way out if you change your mind about a cage you already placed.

Nothing in a fight. It has almost no health, nothing in the world can hurt it except you, and it neither helps nor hinders anything. It is scenery with a spawn condition attached.