You ask several questions --

<< Can we head her off by finalizing quickly?>>
Your Husband's marital status isn't going to affect his child support obligation, which will be based on his income, the number of overnights he has with the child, and several other factors.

The only effect will be whether he receives the "Other Dependent Deduction" credit for the children he has with you or receives the "Prior Child Support Orders" credit. He should discuss with his attorney which is better - you will certainly want to make sure that any child support he is paying to you is reflected in the order (you could probably legitimately do things like working out his college contribution and including it in the order - the higher his obligation to you, the lower the support obligation to the viper).

Additionally, if he's paying you alimony, the court will deduct that from his income before calculating his support obligation to the viper.

<< Can she tap into my support later if she is in fact pregnant with his child?>>
No.

<< Can I sue her since she was the ultimate cause of divorce?>>
Nope. In some states you can, but NJ enacted the "Heart Balm Act" in the early 1970's explicitly removing this cause of action in NJ.

Once the child is born, if she is a "pro" she will probably seek child support from him which he will be required to pay. However, any amounts that are previously set for alimony and child support would not be effected by a subsequent child support award. He would have to make an application to reduce his child support to your children because of the new child support award. I can't tell from your description if he would ask to reduce your child support. If possible, you sould resolve your alimony and child support issues before the new child is born.
As for the other worman, she cannot touch your money. Once it is yours, it is yours.
You cannot sue her for causing the divorce, however, if it turns out he gave her money during the affair, you could sue her to get it back, but it would be hard. You would be better off trying to get it back from your Husband's share of any assets.
DISCLAIMER: This answer is provided for informational purposes only. It is not intended to form any kind of attorney/client relationship.

You are asking interesting questions. A new child to your soon to be ex husband will not affect alimony. It may impact child support because those funds are spread among all children of your soon to be ex husband. She cannot come after you at all and you cannot "sue" her for divorce. You could seek to amend your divorce pleading to add adultery and possibly seek repayment to me if your soon to be ex husband used marital funds to further any affair.

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I am a Washington attorney and have no knowledge regarding the laws, rules or procedures in the State of NJ. If your matter was taking place here in Washington, the following would be my answer:

The soonest you can get divorced in this State is 90 days. The 90 days is known as the "cooling off" period. Obviously, if you can get this divorce filed and completed before the child is born and the mother's files a paternity and child support action, that may be advantageous to you financially, depending on the totality of the divorce settlement, of course. She can still go after him for child support after your divorce is finalized but, presumably, she will be paid less because he'll already be paying child support for two children. This is a no fault divorce state so there's no lawsuit that I know of that can be initiated against a "homewrecker."

I highly recommend you confer with an experienced divorce attorney in your state regarding this matter.

DISCLAIMER: This answer is provided for general educational purposes only. By using or participating in this site you agree and understand that there is no attorney client relationship between you and the attorney responding, and no attorney-client confidentiality. The law changes frequently, and varies from jurisdiction to jurisdiction. The information provided in this answer is general in nature and may not apply to the factual circumstances described in your question. The applicable law and the appropriate answer may be different in the State or States where the relevant facts occurred. For a definitive answer you should seek legal advice from an attorney who (1) is licensed to practice in the state which has jurisdiction; (2) has experience in the area of law you are asking about, and (3) has been retained as your attorney for representation or consultation. Your question and the attorney’s answer may be used for promotional or educational purposes.