Whether an estate is required to go through probate or not, California law provides that a deceased person's will is supposed to be lodged in the probate court within 30 days after death. So yes, upon your mother's death, you will need to lodge the will with the superior court in the county where she lived.
As to what your sibling will need to claim her share of the estate - the insurance company will probably want a certified death certificate and a claim form. The claim form will probably need to be notarized. Each insurance company has its own procedure, so what they'll want depends on which company is involved.
The bank will definitely require an Affidavit of Small Estates and a certified copy of the death certificate. Hopefully, you will deal with someone at the bank who knows what s/he is doing - otherwise, you'll be beating your head (or your sister will be beating her head) against the wall.
In order to open an estate account, most banks require letters testamentary - which you want to avoid.
As for whether you need an inventory, it certainly wouldn't hurt, but it doesn't sound like it's required based on the information you've provided here.
The information you obtain at this site is not, nor is it intended to be, legal advice. You should consult an attorney for individual advice regarding your own situation.
Yes, you are required to file the will in the CA county in which she died. Since the estate is under $100,000, a probate will not be required and you can use the affidavit process to transfer the assets listed in her will.
Any individual seeking legal advice for their own situation should retain their own legal counsel as this response provides information that is general in nature and not specific to any person's unique situation. Circular 230 Disclaimer - Advice given in this response cannot be used to eliminate penalties with the IRS or any other governmental agency.