Question: Does a sacrifice count, if we obtain the animal without spending money on it, for example if the animal is a present?
You do not have to have spent money on the sacrificial animal.
If you receive an animal as a present , it becomes your possession. You may choose to sacrifice it or not. If you do, you are counted to have sacrificed properly, gaining the sawab for your sacrifice. The person who presented you with the animal also gains sawab for his act.