“My mortgage company may go under. Who owns the house in that instance? The mortgage company’s creditors? How will it affect me? I am all paid up and in good standing. Should I refinance now or not?”
You own the house, as you always have; all the mortgage company has is a lien against the house which it can enforce if you don’t meet your obligations.
When a mortgage company goes under, its mortgages are assets which go to someone. Usually, they are sold to another company and the cash used to pay the first mortgage company’s creditors. This can be a little confusing for the homeowners, but you should get notice of where and to whom you need to send your payments. At a minimum, you should keep especially careful records of your payments during this transition period, just in case there’s a dispute later.
The rest of the terms of your loan are governed by the documents you signed, and should not change.