No, you would not be responsible. Your husband's debts are his debts. Indirectly you would be affected by his support because it takes money out of your household. You will not be responsible for this debt if he stops paying.
No, you wil not be responsible for your new husband's child support or his debts. However, if he has an arearage in child support, it is likely that the Child Support Enforcement Agency will file for a tax refund intercept. If you file joint tax returns, the agency will get your portion of the refund along with his refund. Therefore, you would be well advised to file separate tax returns to avoid this problem. If you file a separate return, the agency cannot intercept your refund. You should consult with an attorney to explor all the options.
No you cannot be held responsible for his support obligation directly.
This answer is provided for general purposes only. If you need legal assistance you should consult with an attorney. Responding to questions DOES NOT create an attorney-client relationship.
No, by simply choosing to marry him, your fiance alone will be responsible for these payments. You are not the biological mother. Prudent question to ask though, and good luck!