Your property and income are not at risk UNLESS you and your boyfriend have bank accounts or other property held with both your and his name on them, or you're receiving payments which are clear intedned clearly for him (if, for instance, he's working, but his employer is issuing checks to you).
There's no common-law marriage in California, and there's no "common-law domestic partnership" anywhere.
On the other hand, unpaid California money judgments, including dissolution judgments, bear interest at 10% per year until paid, and judgments from dissolution proceedings are, in general, not dischargeable in bankruptcy, so this is something your boyfriend needs to take care of, sooner rather than later, because it's otherwise going to be an ever-larger problem.
In order to be consider domestic partners in the State of California, you must register with the secretary of state and fill out the proper form. If you have not done so, your domsetic partnership is not recognized by California and there are no community property rights or accrual. She is simply make threats at this point.
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Just cohabitating with someone doesn't make you a domestic partner, which requires a same sex relationship or a relationship between opposite sex partners in which one partner is at least 62 years old. a CA domestic partnership requires registry of the CA Secretary of State (please see the link below).
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.
Please try to ignore your boyfriend's ex-wife, who is either trying to upset you or does not fully understand the law in this area. You are neither your boyfriend's spouse or domestic partner. Domestic partners are same-sex partners or over the age of 62, and you therefore don't legally meet the requirements.
If she wants spousal or child support and your boyfriend has no income to pay it to her, you might want to consider consulting with an attorney about modifying the support order(s). So long as the current order stands, the arrears will keep adding up, and your boyfriend will have no relief until he does something about it. The earliest date the Court can modify your order is the date he files something.