What legal rights do I have to retrieve property from my legal residence even though I don't own the house?

This information should be helpful: https://nycourts.gov/courts/nyc/housing/lockouts.shtml

You should not make unilateral decisions regarding the amount of your child support obligation or the manner in which you pay it. You should follow...

When i file I-751 can I get the new GC with the new name?

You should post your question in the immigration law forum: http://www.avvo.com/topics/green-cards

How do I reopen a family court order?

I would advise you to follow up with your attorney, discuss this issue with him/her and request that a conference call be scheduled with the court....

What do I need to do to file for a quick divorce in New York state?

You will find a lot of useful information here: https://www.nycourts.gov/divorce/divorce_withchildrenunder21.shtml

Can she arbitrarily make these decisions legally?

It is not clear what decisions you are referring to. It looks like your ex-wife manipulates your children by providing them false information...

How does one "settle a judgment" in a contested divorce in NYS Supreme Court?

You may find this useful: https://www.nycourts.gov/courts/6jd/forms/SRForms/orders_proced.pdf

Can I be arrested for taking my kids on family vacation for 10 days?

You are bringing up several issues - parental alienation and parenting time/visitation. Re the vacation, you need to carefully review the terms of...

Can I get a divorce judge to award me attorneys fees BEFORE I hire an attorney?

I don't think the judge will grant your request. An application for counsel fees must contain certain information. For example, it must refer to...

CAN I I GIVE IT DIRECTLY TO MY DAUGHTER?

Child support must be paid in accordance with the court order or the settlement agreement. If you choose a different method, even if you believe it...