Legal Question in Family Law in California

You cannot assert the 5th Amendment to avoid testifying entirely. That only applies to criminal trials. You may however depending on the situation have the right to assert the Fifth Amendment as to any particular question during the trial. The issue is whether answering the question truthfully would implicate you in a crime. So, for example if there is a domestic violence component to the trial you may be allowed to plead the fifth amendment as to that. Also if you have committed financial crimes and answering questions regarding your community property and assets would implicate you in those crimes you might possibly be able to assert the Fifth Amendment in that situation. Other than domestic violence situations, however, it is extremely rare for there to be a family law situation in which the Fifth Amendment would properly be asserted. Trying to do it without the advice of counsel would be a huge mistake.

Sure if it is regards to admitting a criminal matter. If the criminal matter reflects on your parenting or hiding money, be prepared to lose....

Only as to questions for which the answer would truly incriminate you. You cannot assert the fifth simply because you don't want someone else to know the answer to the question.