If my mom died intestate with a mortgage loan on the house, I want what is fair to all three of us, what are our options?

Assuming your father is dead, in California the property would be divided into thirds, subject to the mortgage. Most bank loans have a provision that if the borrower dies, the loan can be called and made immediately due. So you first need to check to see what the bank is willing to do. If your sister has enough money to qualify for the loan [which I doubt as otherwise she would be living in her own home], they will probably allow her to take it over, but may charge some extra fees for the switch. Otherwise, you would probably have to sell it to pay off the mortgage and then split the profit three ways. Legally, all three of you are equally liable for the monthly mortgage payment. For ownership to pass, the estate must go through probate and during that time the administer of the estate is entitled to require your sister to leave or pay reasonable rent to the estate. Once the property three, none of the owners can be charged rent by the other owners. You would have the right to live there rent free. The fair way of handling it would be that she pays the entire monthly mortgage herself and the proceeds from the sale are divided into equal thirds with her keeping the house [on paper, she is getting much more as she keeps the house, but she is also assuming the debt for the house so the net result is the money assets are being divided equally]. That she has lived there rent free for 13 years does not give her any greater legal or equitable rights than your brother and you.

Unfortunately, you as heirs don't get to decide what is fair. The state determines the matter in probate.