Translation:She gives him a mouse, and gives us a cat.
The unstressed pronouns is always BEFORE and NEXT TO the verb and the stressed pronoun can be AFTER the verb or in order to emphase more you can place it in the first place.
You do not give us a dog.
(Tu) nu ne dai un câine.
Nu ne dai nouă un câine.
Even more emphasis
You do not give a dog to us but to him
Nouă nu ne dai un câine ci lui.
Two questions: 1. Isn't "îi" for plural? How does this fit in here? 2. Does the object always get two pronouns? ("us" here seems to have two: "nouă" and "ne"). There seem to be a lot of pronouns to choose from and I'm getting very confused.
I'm not a native speaker, but I can answer some of your questions.
Îi is in the dative case, third person singular.
I think you can leave the stressed forms (lui & nouă in this case) out.
You certainly need the two pronouns in the first clause. "Îi" only tells us that someone is getting a mouse; it takes the extra 'lui' to clarify that it's a him getting, and not a her. The extra pronoun in the second clause is for emphasis, and to show a contrast between what the two parties got. "...and gives /us/ a cat".
What i can't answer is why the pronouns are positioned differently....