Can I get sole custody

You would most likely get temporary primary custody based on the child's immediate needs at this point in time. If he does not agree there would...

Boarding a flight with a domestic violence charge.

Was an arrest warrant ever issued for him and were charges ever filed against him?

The signature portion can be in blue ink. The actual document cannot be in blue ink and only black ink.

Could I be fined?

This is not a child custody matter. I will re-post to the proper forum.

Can we go back to court for equal visitation, 50/50 time, at this point after so many years of an existing arrangement?

There needs to be a substantial change in circumstances for the court to order a change of that nature.

What can I do legally for compensation from my greedy cousins?Did They have a right to change locks, trespass, and remove items

You need to consult with an attorney. If there was undue influence exerted on your relative then you need to petition the court.

Is the divorce valid

New York does recognize foreign divorces.

What to do when the mother violate the Custody Order by not releasing my daughter and creating troubles?

You should speak to an attorney. You can file an enforcement petition and request a change in custody because of her actions.

Since he's already signed the Acknowledgemt of Paternity can we still get a paternity test through court?

He can pay privately and get the test done.

Would it be illegal to ask a legal father to babysit his child ? is that harassment, the parents are no longer together ?

It is not legal to ask a father to care for his child. It would not be considered harassment unless asked In some manner that was harassing.