Yes, it is important to check the serial number of the item you are returning for a refund.
Yes, most stores keep track of the serial numbers of the products they sell. This is for security purposes so that if they are stolen or lost, the store can identify them and refuse to accept them as a return.
There are a number of websites that can help you check if the serial number is stolen. You should be able to find one with a quick internet search.
Walmart does not typically check the serial numbers of products.