You should consult with an immigration attorney to determine whether you qualify to sponsor him for permanent residency. These cases can be complex and your husband may or may not be eligible to adjust his status from inside the United States.
As long as you maintain your H4 visa status throughout the asylum process, you will not lose it even if asylum is denied. Asylum cases are difficult to win, you should retain an immigration attorney to represent you.
You can only adjust status if you have an approved I-130 and a current priority date. You did not mention any cases filed for you, so you and your mother need to schedule a consultation with an immigration attorney to plan a course of action for you. Do not overstay your visa admission.
It's not clear whether you have removed the conditions on your green card or not. You should consult with an immigration attorney to determine if and when you will be eligible to naturalize based on your marital situation.
Whether you qualify for any of the investor visa programs will depend on the amount of investment, your nationality, the nationality of the business, and other factors. If your parents are looking to open a subsidiary of their company in the United States, that could open up additional immigration paths for your, if you work with them. Consult an immigration familiar with investor visas when you are ready to proceed. (Avvo is not the right place to discuss your financial and immigration...