Does HR have the right to file a 8x10 naked picture of me in my personnel file when it is not locked or kept secured ?

Id argue that it's not work related, either the transmission or need for retention--- and must be purged from the file.

I am the petitioner and I missed the sixty days filing of my case management order on the respondent. what do I do? any fines?

File it now and I doubt you'll hear about it again. If you do, submit a response exhaling why it was late.

What are the chances of me being the primary custodial parent of my child?

Likely a "step up" visitation plan will be ordered.

A 730 evaluation with psychological testing

My ex wife has custody of my son & daughter (twins). Her new boyfriend was charged with Domestic Violence, assault, battery.

File a Request for Orders, noticed motion not an ex parte. If not resolved thru mediation it is likely a 730 evaluation will be ordered.

Do the regular every other weekend visitation change for Holiday scheduled weekends.

Generally, the holiday schedule supersedes the regular schedule.

Domestic battery charge not being prosecuted by the DA, is this still valuable admissible evidence in family law and mediation?

The answer is "yes" you can still present the evidence in family court and the burden of proof is less than in a criminal case.

I work for a law enforcement communications center in California. I was asked by my manager to do do a follow up on a work comp

You are supposed to get permission before you record the call, not at the end.

MSC in Riverside self-lit dissolution requested/scheduled by respondent too early. No way to be ready to discuss settlement.

You can show up and request more time.

Can ex take minor child around person if there is a restraining order on person protecting minor child?

No. Consider enforcement of restraining order and restrictions on visitations.