You don't say what the finally outcome of the domestic violence charge was. If is possible for an arrest for domestic violence and the DWI to show up in a background check. I don't think there is a way to expunge either of these if they were adjudicated and she took probation. Your mother needs to be careful because the Nursing Board could consider this a pattern of alcohol abuse causing her to be an impaired nurse. This could lead to the loss of her license.
In addition to Ms. Trevino's answer, the law is that person's with any level assault cannot work with children or the elderly - even as a volunteer! Supposedly this is for the protection of the helpless. If your mother received deferred adjudication and served out the probation, she can move to essentially seal the record (non-disclosure) but the record remains open to law enforcement (which likely includes the licensing division.)
As far as the DWI, if she was convicted (the case was not dismissed because one cannot receive deferred adjudication on a DWI), then it will remain on her record indefinitely.
I suggest, as far as your mother's license, that she get herself into a treatment program immediately to demonstrate to the board that she has recognized a potential problem and is taking care of it without being told to do so (which will demonstrate a truer interest in healing herself.)