Can the owner of the salon I rent a booth from bill me for a portion of her business insurance? No contract, out of the blue

This is a very good question, but this is also highly dependent on course of business practices between you and the salon owner.
Booth rentals are a unique thing in CA and have been regulated and taxed in different ways. I've seen situations where a salon owner has collected "rents" that included commissions, etc. and later on been determined to be an employer/employee rather than free-agent or sub-contractor situation.
You should consider talking to an attorney that is familiar with labor/employment law in your area and who handles salon related issues. Generally there are a number of state-wide and local licensing requirements, and among them you need public liability insurance for the premises, and your activities.
Click on "Find a Lawyer" above and locate an attorney who handles labor/employment law issues. I would also suggest considering written agreements instead of oral agreements.

The answer to your question will depend on whether you are properly characterized as an independent contractor truly renting the booth versus a mischaracterized employee. As an independent contractor, what the owner of the salon decides to charge you for rent is completely subject to your contract with the owner. If you have a month-to-month rental then the salon owner can change the terms and conditions of your rental at the beginning of any new month, just like a raise in the rent. There is no law that prevents contracting parties from agreeing to pay for insurance as part of the rent.
However, the salon industry is rife with mischaracterization issues where employees are wrongly labeled independent contractors. If you are in reality an employee, an employer cannot foist off on you part of the overhead. If you think you have been mischaracterized, that is the mere tip of the iceberg in terms of violations of the Labor Code and it would be prudent for you to seek the confidential advice of an employment attorney if you think that is a possibility.
Good luck to you.