You get the Snake token no matter who controlled the target creature.

Rite of the Serpent won’t create more than one Snake token, even if the target creature had more than one +1/+1 counter on it.

If the creature is an illegal target as Rite of the Serpent tries to resolve, Rite of the Serpent won’t resolve and none of its effects will occur. You won’t get a Snake token.

If Rite of the Serpent resolves but the creature isn’t destroyed (perhaps because it has indestructible or it regenerated), you’ll get a Snake token if the creature has a +1/+1 counter on it.