If you are no longer on probation and have only one conviction for a simple DUI, it probably will not prevent you from establishing 5 years' good moral character required for naturalization, but you should definitely consult with an attorney to discuss your case further.
Domestic Violence convictions are removable offenses and your boyfriend appears to be subject to mandatory detention. Please consult with an immigration attorney regarding strategies for relief from removal.
If your father was previously deported and reentered the US unlawfully, his deportation can be reinstated without further proceedings before an immigration judge. Consult with a US Immigration and nationality law attorney as soon as possible to determine whether removal can be avoided.
If you are a US Citizen and at least 21 years of age, you may petition your mother to immigrate to the US. You need to hire an immigration attorney to clarify any admissibility issues for you directly.