No. CA doesn't recognize "common law" marriages based on living together, or having kids together, or thinking of yourselves as married and holding yourselves out as married. Either you've gotten a marriage license and had it filed, or you're not marriied under CA law for any purpose, including tax status.
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.
No, you can only file a joint tax return if you are married. You are not married and California does not recognize common law marriages.
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.