Generally, you need your spouse to help you petition for your green card. If you are no longer together, then that door is closed. However, there may be a way to move forward without your spouse if you were subject to any sort of abuse or cruelty. It's a much more difficult way to get your green card, but if it's an option for you, then it will certainly help. Please, speak with an attorney to see if anything else can be done.
I would talk with a criminal defense attorney to see if there is a way to get your domestic violence sentence modified, clarified, or if it's possible for it to go away entirely. If you can get the criminal issue cleared-up, an experienced immigration attorney may be able to help you with your status in the US.
No, not all J-1 programs are subject to the two-year home residency requirement. It depends on 1) your program type, 2) your country of citizenship, and 3) how your program is funded. You and your employer should speak with an attorney to review your options.
While you can file for asylum, you will have an *extremely* difficult time winning. You have to file for asylum within one year of entering the United States. If you file afterwards, you have to either show 1) changed country conditions or 2) exceptional circumstances. Depending on where you are from, it may be difficult to show option 1. Under option 2, you should have filed for asylum right after your daughter was born in order to have a stronger case. Filing for asylum is difficult and the...
Your step-daughter would have to come to the US as a tourist. The problem is that she will likely be denied a tourist visa because she has so many family ties in the United States. It's not impossible, but she'll really have to work hard to show she has an intent to return to her home country (other family, finances, job, etc.).