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.
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....
1) îi is both the accusative plural and the dative masculine singular. It is like many other words in different languages that are spelled the same but have different uses. You can tell which one it is based on the usage and the verb that it is attached to. If it is a verb like 'to give' then îi would mean 'to him', while if the verb is 'to love' then îi would mean 'them'. You unfortunately just have to memorize which pronoun case matches with which verb as some of the ways in which the verbs are conceived in Romanian are not the way a native English speaker would comprehend them.
2) The other response is incorrect, you do NOT always need two pronouns. You include the stressed version optionally for emphasis or clarity, and the different positions are also for emphasis.