If I place a few toys in front of a child, these are this kid's toys, he knows them, and then I put these toys in a bag and I want to know if the kid can remember which toys I put in the bag (testing his memory), when I say 

- What's in the bag?
can the kid say
- There's THE mouse and THE cat in the bag.
Is it fine to use THE after "there is" in this situation? I suppose it's fine since the kid knows the toys.
When you begin the sentence with "There's" or "There are", the definite article "the" sounds awkward.  Therefore, the child could say, "There are A mouse and A cat in the bag."  However, since the specific toy mouse and the specific toy cat were previously shown to the child before they were put in the bag, he could also say, "The mouse and the cat are in the bag."  

- what's in the bag?
- The cat is in the bag.
I'm telling my kids that "the cat is in the bag" is the answer to a different question, i,e, "Where is the cat?". and if the question starts with "What's in..." then the full answer should always start with "There is/are..." am I right?   "There are a mouse and cat in the bag" is correct.  However, as we saw before, because the child saw the cat before it was put into the bag, it is a "specific" toy to him.  Therefore, he could also say, "The cat is in the bag."

It may be interesting to know that the idiom "It's in the bag" means that something is as good as done; certain to be realized.   "The candidate of Party X is 20 points ahead in the polls.  It looks like the election's in the bag for him."   And the idiom "to let the cat out of the bag" means to reveal a secret.   "It was supposed to be a surprise party for Judy, but someone let the cat out of the bag, and she found out beforehand."  Variant:  "It was supposed to be a surprise party for Judy, but the cat's out of the bag.  Judy knows all about it."