You can and should file taxes as married. Even if your husband still doesn't have a SSN, he may request a tax ID number from the IRS. Filing as married will also help with your application. Generally speaking, you should only claim as head of household if you're single.
[This answer is for general purposes only; it does not constitute advice and does not establish an attorney-client relationship.]
If he is eligible for adjustment he can get a work permit and social security. Regardless you may file as married since you are in fact married.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.
Yes, you file as Married.
Evan A. Nielsen is licensed to practice law in California and handles federal tax matters throughout the U.S. The information provided here is for educational purposes only and is not intended as legal advice for a particular matter. This response does not create any attorney-client relationship with the author. For specific advice about your particular situation, please consult an attorney.