Her legal status should not matter as to whether or not she recieves child support. However, your state in particular your sherrif has a history of being very aggressive with immigration type issues. I would file an action in your court and if you like, seek custody or visitation, but please work with a local attny, you do not want her running out of the US with the kids, in the interim , have the court or counsel hold the passports etc, take care.
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I agree with attorney Lewis. The sooner you take action, the better. Consult with a family law attorney as soon as possible and file for divorce and temporary orders, to allow you to see your children. If you believe there is a threat your wife may take the children out of the country, you may also need to file for emergency orders to prevent that from happening.
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The immigration issue probably won't affect the court's decision regarding custody; if it does it will only be one factor among many that the court considers. If she has filed for custody and child support, be sure to respond to her petition and defend your case effectively. Courts are strongly in favor of joint custody (they call it legal decision-making) and substantially equal parenting time.
The child support guidelines are incredibly complex and I would recommend getting an attorney in AZ to help you through the process. You may end up paying tens of thousands of dollars in child support that you shouldn't have been ordered to pay in the first place. After child support becomes past due it is effectively impossible to get rid of that debt without an agreement from the other party.
This post is general information about divorce and family law in Arizona, and is not legal advice. Divorce is a complex process, and taking legal action without the one-on-one professional advice of a lawyer can produce unexpected results.