Is this acceptable if the real property is physically located in CA? Yes.
Are we required to transfer the mortgage on this home to the LLC as it is already in both of our names? If you want the mortgage to be an LLC asset, then you need to transfer it to the LLC in exchange for membership interests in the LLC.
A Washington LLC can hold property in California. Be aware, though, that California law will apply to any activity related to the property while Washington law will apply to the LLC. You will likely be taxed in Washington, which is probably preferable in that California taxes tend to be higher.
Your mortgagee, however, may not allow you to transfer tjhe property to the LLC, which defeats the purpose of creating it generally. Most mortgages contain a due on sale clause, which provides that any transfer of the property requires repayment of the mortgage in full. You should check with your mortgagee before doing anything.
I have had this issue with a couple of clients and the banks or other mortgagees have generally prohibited the transfer of the property. You also need to be certain that there are no transfer or recoding tases that would strongly impact the transfer. Although I think it is somewhat unique, in Maryland, for example, if owners transfger to an LLC it is a taxable event and there is a substantial tax due on the transfer.
Yes, a Washington LLC may own property in California, but as it will be conducting business in California, it will need to register in California as a foreign business entity. You brother can probably be the registered agent, but you really need to talk to a California attorney about it.
As Mr. Sprang notes, beware the due on sale clause. You need to talk to your mortgage holder before you convey. At the least, your mortgage holder will want a new note and security document and probably want you and your brother to sign personal guarantees before it lets you transfer. Make sure the guarantees don't open you up to general liability.