Ok. Understood. There are a number of issues here.
Since your husband is a nonresident alien, you would normally have two options. First, you could treat your husband as a resident alien for tax purposes and file a joint tax return. You would get an exemption for your husband, but all of his worldwide income will be taxed by the United States. The problem is that he's been using a false SSN. That means that his employer has been withholding tax, and paying over to the U.S., though because he's using a false SSN, that withholding has not been credited to him. Your choice would be to come clean and admit to using a false SSN, and claiming the taxes reported on his W-2, or lying on the return and saying he had no income.
Your other choice is to treat your spouse as a nonresident alien for tax purposes. If you choose this option, you cannot file a joint tax return. You must file with a status of "married filing separately." You normally can only claim a spousal exemption of your spouse has no U.S. income, but yours does, even if not properly reported.
The problem you'll run into with this option is, like above, if you use his ITIN as an identifier, it will be different from the false SSN on his W-2.
The above said, courts have recently sided with nonresident aliens that have, for example, used someone else's SSN to get a loan, or purchase a car. Thus, the greater of the two dangers is evading taxation by the U.S. You may want to consider filing a joint return, treating him as a resident alien, claiming the exemption, and reporting his W-2 income on the joint return.
While this is not ideal, and could present some immigration problems, at least it will not be tax evasion. Additionally, you need to consider that of you're taking steps to assist him in not reporting federal tax, you may also be guilty of evading taxation.
If you own the house, you'll qualify for the normal deductions, property tax, mortgage interest, etc.
You cannot claim his children as dependents for a deduction. They do not live with him/you, I'm assuming do not have ITIN s, and do not live in the U.S.