You can get an order, but you will need his last name. Collecting may be difficult if he stays in Mexico.
Several things are not clear from your question. If this is a Michigan order, Michigan law applies and you will have to ask lawyers who practice in Michigan. If this is an Arizona temporary order, the other parent does not get to take a child out of state while the case is pending without a court order or a written agreement from the other parent. Also, it sounds like the other parent may be planning on violating the temporary order. Also, if the order is that the other parent does get the child in Arizona for two months, it may be possible to change the temporary order on the grounds that the child will miss two months of school.
The court will appoint a lawyer to represent you.
Yes, no one needs his or her spouse's consent to get a divorce. No one needs the other spouse's signature to get a divorce.
She can get an orde rof protection against him if she has a valid reason to do so. this would require him to leave immediately. If she doesn't have a reason to get an order of protection, she will hav eto follow the eviction process, which involves givign him notice to leave, waiting so many days, and then filing in court. Before any of that, she might want to just ask him to leave.
To get an annulment, you have to show that the marriage was invalid from its inception. the reasons include fraud, polygamy, no license, and the like.
DCS has immunity for mos tthings. You did not list anything actionable in your question.
This pattern will not help him in court, but it has only been one week. You cannot force the other parent to see the child or ask about the child. You can force the other parent to pay child support by getting a court order.
To answer your question, no. However, you can file an enforcement action to make sure he pays the current amount no matter what, get a judgment for the arrears, and then gaarnish his pay at his second job to start satisfying the judgment.
DCS has immunity for the things you mention in your question. If you're going to sue them you will hav eto find something else like defamation or something else that is not directly related to doing their job. ONe family tried this about ten years ago and it ultimately failed.