Depends. Hire an attorney for optimum results.

Contact the court clerk or an attorney to see what's going on.

What is the consequence if a 36 year old woman has sex with a 16 year old male

16 is the age of consent unless there is a teacher involved.

Can a current DSS case In oaklahoma be moved to California

It depends. Hire an attorney for optimum results.

Can I own or buy a firearm?

You may need an expungement. However, it does not appear as if you are a convicted felon.

They will probably be tried as a juvenile but they could try them as an adult but I doubt it.

Can I legally move myself and my two sons out of the marital home and into an apartment down the street?

You need to file a divorce and the divorce should involve your desire to move.

Can I file a motion to dismiss?

Find another attorney. I doubt you would be successful on a Motion to Dismiss based on what you have stated herein.

What if my soon to be ex husband won't sign the denial papers? Does he have a window in which to sign them?

Seek an attorney as there are too many unanswered questions here. It could be either 60 days or 2 years.

What's the most likely result for custody? Should I speak to a lawyer?

Yes you should get a lawyer. Doing this yourself is never ever a good idea.