First, you could file for enforcement of the terms of the current custody order. Additionally, in joint custody situations like yours where the parties cannot seem to communicate, you could ask that the court appoint a "parenting coordinator". For more information on parenting coordinators, I would direct you to the Arizona Family Law Blog on my website.
I actually practive family law appeals in the state of Arizona. The docketing statement must include a list of what you believe to be your arguable issues on appeal. There are specific timelines for the whole appeal process. You must have also filed your "notice of appeal" within 30 days after the date of the judgment you seek to appeal. You must also ensure that you have ordered all the appropriate briefs to be used in your opening brief. If interestes, please contact me for a free...
You could seek an injunction against harassment against her or in the alternative if timely; request a hearing on her order of protection wherein you may present evidence to the court of her actions in hopes that the order will be quashed.
A default judgment can be set aside pursuant to Arizona Rules of Family Law Procedure (ARFLP) rule 45 (C). The rule states that "for good cause shown the court may set aside an entry of default and, if a judgment has been entered, may likewise set it aside accordance with Rule 85 (C). As long as what you are stating is true, you can file a motion with the court and argue that from Petitioner's actions good cause is present to set aside the default judgment.
Check with an attorney in your state for further clarification, but usually the parent seeking to move must receive written permission from the other parent prior to moving or request such permission from the court.
I would suggest contacting an immigration attorney to discuss what will happen to your husband's pending immigration statues in the event you do divorce. I presume that an I-130, adjustment of status was filed by you on his behalf, but an immigration attorney is better fit to answer this question.
Child support orders can only be modified in states other than where the original order was issued in very limited circumstances. I would suggest filing a motion to dismiss with the state court where the new action was filed based upon the fact that the new state lacks jurisdiction to establish a new child support order.