According to the information currently available, and this can change, the USCIS will continue most operations because it is funded by fees, not tax dollars. Other immigration agencies, however, will shut down completely or partially. If your attorney does not know this, you need to find a new attorney.
If the attorney helping you and your employer through this process cannot answer these questions, I recommend that you find a new attorney. If you do not have an attorney helping you, I recommend that you speak to one. As you are finding out, this process is not as simple as it might seem and there are many areas where having an experienced attorney will help you and your family. The process and your questions are complex and cannot be answered in detail except by an attorney who has been...
Even arrests or convictions for relatively minor offenses can have serious immigration consequences. You should speak with an immigration attorney experienced in representing noncitizens with a criminal record.
It is not clear that you are eligible for adjustment of status, so you need to have an immigration law attorney review your situation before you do anything. Also, you may be eligible for other relief that would allow you to live and work in the U.S., at least temporarily, until you are eligible for apply for a green card or other permanent status.
You already have an attorney and should direct your questions to him or her. After all, that is what the attorney is being paid for. If you do not have confidence in this attorney, you should hire your own independent counsel or perhaps you and your company should find a new attorney.
An L2 is for a dependent of an L1. If the L1 no longer holds that status, your L2 status is no longer valid, so you would either need to obtain a new nonimmigrant status (perhaps an H4) or leave the U.S. L2's can get EADs but H4's cannot. If you want to continue to work, you need to find a status that would allow you to do so. You and your husband should speak with his employer and attorney to see if it is really necessary for him to change to H-1B status.
An immigration attorney can help you find out why he cannot enter and what steps he needs to take to get permission to enter. He may need a waiver of inadmissibility for some past immigration or criminal violation.
Telling you what forms to complete or how to complete them is legal advice that should only be provided by an attorney with whom you have an attorney-client relationship. Advice given without a full understanding of the facts can have adverse consequences for you and your family.