The statute of limitations for medical malpractice is three years from injury or one year from discovery (or when the plaintiff, exercising reasonable diligence, should have discovered the injury). Code of Civil Procedure, Sec. 340.5.
I am unaware of any separate statute of limitations for a class action. It depends on the basis of the claim.
A class action requires common questions of fact. Off hand, I doubt that medical malpractice case would satisfy that requirement. Talk with a med-mal specialist (not us).
The State of CA would not be liable for "negligent licensing," as there is no such claim. Depending on what waivers and releases you signed, you may still be able to sue the doctor.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.