My child's father wants to give up his parental rights to avoid child support

It is t possible to do that. Don’t give up your arrears.

Do I have to pay temporary spousal support while appeal is pending?

Yes. The order is in effect and if you do not pay you are violating a court order.

Supreme court order on child support and custody and visitation order can be tried in family court as circumstances changed?

You need to amend the judgement of divorce.

Under what circumstances can a child abuse case be brought against parents.

By whom? The child? Or ACS? ACS can bring when they have evidence of possible abuse or neglect.

Child support

Who is on the birth certificate? This is sticky since as her spouses he is the presumptive parent.

Can my husband kick me and our children out of my in-laws' home?

No. He cannot do that if he is abusing you call the police and press charges and get a restraining order.

How can I get a divorce and alimony, child support from someone who's been deported to their native country?

There is no way for the courts here to enforce any order entered. If you do get an order here will Greece give it comity and enforce it on him there?

I have a charge of harassment in the second degree. What is the punishment if I plead guilty?

Don’t take a plea. Fight it. A guilty plea will never leave your record and will affect your work future negatively.

If the family court judge denies your objection then you would need to file an appeal. If you do not have the funds for an attorney for the appeal...

How can I enforce visitations?

File the petition in family court to enforce the court ordered visits.