As an immigration expert with many asylum cases filed in Miami, I can respond clearly: YES. There are many delays recently. Please remember that applicants should apply for a work permit 150 days after filing the asylum application.
For other information about asylum, see my blog: http://asyleesdaughter.blogspot.com/.
Regulations allow applicants to file Form N-400 three months before the fifth anniversary. So, if you are eligible on October 2015, you can apply in July 2015.
To be certain that your case has the highest likelihood of approval, consult with an immigration attorney.
Foreigners need to prove they marry with the intent to build a life together. That is difficult to do, as you are seeing, when the U.S. citizen "changes" after the marriage. Yours is not an easy case to win based on the little evidence you say you have. I suggest you meet with an experienced attorney who will guide you as to how to organize evidence, and even represent you at an interview if needed.
Obtaining a second citizenship, by itself, may not be a reason to lose your green card. The reason why you need that second citizenship may be something they will look into, however if you are looking at living in a second country for an extended period of time, the immigration officers may ask questions to be sure you have not abandoned you residency. Please consult with an experienced immigration lawyer. My blog on asylum is at Asylee's Daughter in blogspot.
An application for asylum does not affect the L1A status. In fact, if you are still in status when the decision on the asylum application is given, it is a benefit to you. Foreigners who are not in status and receive a denial must defend their case with an Immigration Judge. For more information on asylum, please see my blog. For information on scholarships for asylees, see the other link, too.
Divorcing does not affect a VAWA case. Some survivors remain married to the abusers; some do not. To be SURE you will get a green card based on VAWA, go see an experienced attorney. She (or he, if you feel more comfortable with a man) will tell you what you can do now -- before divorcing -- to increase your chance for success. Good luck.
It depends on what you mean by "belligerent and malicious." There are many types of abuse and an experienced immigration attorney can listen to the facts and tell you whether it is possible to get residency based on VAWA. That's the Violence Against Women Act. This author has obtained green cards based on VAWA after one incident of non-physical abuse.
A motion to reopen would allow you to present this new evidence. You must use the experience of a good lawyer because the BIA will require an explanation as to why the petition was not filed. In general, it is difficult to win a motion to reopen.
First, get out of the abusive relationship and take care of yourself and your children. If you are the victim of abuse, there IS a safe alternative. There are places like the Miami Rescue Mission and Safespace. Second, take care of your immigration status. A victim does NOT need a police report to win a VAWA case. The instructions for what documents you should submit with the Form I-360 are available free, as is the form.
However, because you are dealing with something as...