Do the dogs only have a certain amount of time there before they’re euthanized or can they stay however long until they find a home

Only those who cannot be safely trained & placed with family would be euthanized. Even then, the staff & volunteers work with the animals to socialize them. A nursing cat or extremely skittish dog would likely be placed in a (temporary) foster home to minimize distressl, teach good behavior, & get a better idea is their personalities.

Only the animals that come in badly injured etc

I don't believe they euthanize unless necessary.