It depends on the exact wording of the agreement as to when the alimony should start after the separation. Your best option is to speak to a lawyer and show them the agreement. Presuming the agreement states that the alimony payments are separate from the sale of the house, you can seek the court's help in enforcing the agreement.
You are still legally married so it would most likely still be considered marital property. A lawyer might help you negotiate a settlement than can protect the increase. But that is only a possibililty.
There may be other issues besides a home that need to be settled, such as retirement accounts or vehicles. Is your spouse asking for alimony or are you? If there are no issues such as these, the divorce can be quite simple provided you can prove the grounds for the divorce. However, it is unlikely that due to the court scheduling you will be divorced in one month. You must file the papers, the other person must be given a chance to respond, then the court sets the hearing.
Your son can file for custody on his own at the age of 16. Prior to the age of 16, your ex-husband can file for custody provided there has been a material change in circumstances. A material change in circumstances is a change in the parents' or child's lives that affects the current custody arrangement. Such things are a parent moving, the child's grades going down, a parent remarrying or the like. This is not an exhaustive list.
The best way to get your driver's license back is to contact child support enforcement to work out a payment plan. Also, if you are not working, you should file to modify the child support. You should consult an attorney to discuss all your options.
Unless you have a court order giving you primary physical and sole legal custody, you must have the other parent's permision to travel with the child. You can't even get a passport for the child without the other parent's signature.
You should not use your kids as pawns in disagreements with your wife. If she is refusing to give you her new address that is one thing. Refusing to return the children is another thing. Unless you know for sure the children are in immediate danger where she is living, you should return them. Simply not having the address is not enough. You can address the lack of an address in other ways, such as a request to the court.
The scheduling hearing which is set by the court will then determine all future court dates. At that hearing you can ask for a pendente lite hearing (temporary orders). This will set a temporary visitation schedule until there is a final custody hearing. The court is not likely to change the whole custody situation on a temporary basis.
What is actually physical abuse for the purposes of a protective order is governed by the law relating to protective orders. Basically it has to be actual physical touching or threats of imminent physical harm. To get a divorce based on physical abuse has a different standard. If you get a protective order, the mere existence of the order is not enough to prove abuse for a divorce.
As for the effect of the abuse on a pre-nuptial agreement, it depends on the wording of the pre-nuptial...