Legal recourse???

The first thing you should do to protect yourself is to consult a local attorney - as should your fiance! Trying to do this kind of thing on your...

Is it possible for my family to apply for an asylum or immigrate to the US based on domestic violence abroad? (I'm a GC holder)

Consult a local immigration lawyer - and quickly!

If I move in with my boyfriend of 1 year and a half can my ex take my son away ?

You should retain a local family law attorney and discuss the matter in detail and discuss options!

How can we stop these charges, they are not true?

You should speak to his defense attorney and see how he or she thinks the matter can best be resolved.

Can he take my son out of the country without my permission?

Does the child have a passport? Does the father have it? Ordinarily the answer would be no, he can't - at least not legally. But "not legally"...

No. But don't delete the messages! Consult a local attorney and./or the police if the behavior continues.

I want to know if my case has been dismissed

Call the court and ask!

Regarding unreimbursed medicals expenses for the children.

Ordinarily a parent is responsible for children's expenses. Consultation isn't the issue - unless the order under which he is paying says...

California laws on appealinggrandparent visitation rights timeline.

It's not clear exactly what your question is, but you should discuss the situation with a local family law attorney.

File or not seperate from soon to be husband

That's a question you should discuss with your accountant!