If the abuser has been convicted of a domestic violence misdemeanor or felony, can s/he keep or buy a gun?
No. Under federal law, if the abuser has been convicted of a felony or a domestic violence misdemeanor, s/he cannot have or buy a gun.* If you are not sure if the abuser has been convicted of a domestic violence misdemeanor, see What crimes are considered domestic violence misdemeanors? In addition, LA state law says that a conviction for various other crimes would also make it illegal for the abuser to have a gun.** See I am a victim of domestic violence and the abuser has a gun. Is that legal? for a list of those crimes.
* 18 USC § 922(g)(9)
** LA R.S. 14:95.1(A)