Legal Question in Real Estate Law in California

You are required to pay the rent to whomever the owner is. Make sure to keep your rent receipts, canceled checks, etc., so that you can show the bank that forecloses that you're current, and perhaps could work out a continued tenancy or even try to purchase the property. As far as rewriting the lease agreement, you probably would be unprotected if you were to do so. I suspect the landlord would then soon move to evict you -- for what reason, though, I don't know.

Well, you are required to pay rent to someone. It would be the owner up to the point where a lender who has a valid assignment of rents under a deed of trust with assignment of rents comes forward and demands that you pay the lender instead of the defaulting owner.

If the house is sold at a foreclosure sale, your lease will be wiped out unless the loan being foreclosed is within the scope of the new lease-preservation law because of the loan having some federal aspect (FHA, etc.) which is something you probably don't know but might be able to find out. If the new law doesn't apply, you'll be subject to notice and eviction by a buyer at foreclosure.

A good record as a tenant may prompt the new owner to continue to rent to you on favorable terms.