Unless you have grounds, for example marrying a US Citizen, you cannot apply for your green card merely because you are in US.
As my colleague explained the route is a lot longer if your applying for permanent residence based upon employment. There will be many steps that you must complete before you can even file your adjustment of status application.
I do not know your spouse's history, but if your spouse applies for adjustment of status- for example employment based - you can also...
I agree with my colleague, find an attorney that meets your needs and that you feel comfortable with.
The attorney will be able to tell you how they will handle the processing and filing of your case. Also, the preparation for the interview.
Usually, the marriage interview is scheduled for the local office near your home.
Your husband has many things going against him.
Both of you should sit down and have a consultation with an experienced attorney in your area to discuss any options he may have. It is important that the attorney have access to all his immigration and criminal documents.
You can obtain some great attorneys from this site (who also offer free initial consultations) or from www.aila.org.
You should consult an immigration attorney before trying to leave the US or requesting voluntary departure.
The fact that you have children who are US citizens and have never committed any crimes may help you to remain in the US.
Many of the attorneys from this site will give you a free consultation, you should at least give it a try so as not to be separated from your children.
I agree with Nick. Depending on the reason he sought asylum going to his homeland can jeopardize his permanent residence.
If he had an attorney file the asylum case, he should consult with them before making any plans to leave the US and return home.
You should hire an experienced atttorney to handle your case. You want to file the petition as soon as possible. Before USCIS issues a removal/deportation order.
The decision to accept your late filing is discretionaly so it is important to have a strong case and clearly state what led to the delayed filing.
My firm has successfully handled many such cases; as have many of my colleagues on this site.
Please retain an experienced immigration attorney immediately.
Temporary options that could lead to permanent residence:
Employment Visa - H-1B Visa for example. You would need to obtain a sponsor/employer in the US who is willing to file an H-1B visa for you.
There is also an L-1 Visa - Intracompany Transferee - but that would require that there also be a US company to transfer to (subsidary or affiliate of the foreign company).
Immigrant Investor (EB-5) - to promote economic growth in the US through investing in a new commercial enterprise...
Unfortunately, there is very little that she can do, immigration wise.
If she entered the US legally (and has proof of her entry - passport, I-94, Visa) then eventually, her oldest child can sponsor her, when the child is 21 years old. Or if she falls in love with an American, he can file for her adjustment of status.
She can do a search for Free Immigration Legal Services in her area. For instance, if she lives in Missouri, the website would be: http://www.justice.gov/eoir/probono/...