One DUI should not have a negative impact on your application for citizenship. Generally the last five years are reviewed most closely when it comes to criminal activity.
One DUI I does not a drunkard make. As long as your application shows you are generally of good moral character, this one incident should not have a significant impact.
However it might be a good idea to have a experienced immigration lawyer review your application, arrest and court records to determine if there are issues...
There us NO WAY your family member will be re-admitted to the US in a year, if ever. If one does not leave under voluntary departure as promised, s/he is disqualified from most forms of relief for 10 years.
Another consequence of failure to depart the U.S. on time is an automatic ten-year bar from being granted cancellation of removal, waiver, adjustment of status, change of status, registry, and further voluntary departure.
Also a crime involving DV is an offense that would render one...
"Saving time" may not be an option. Your approval receipt will not be adequate to prove citizenship.
You will need copies ( or originals) of your proof of your U.S. citizenship status (a birth certificate, naturalization certificate, or passport) or permanent resident status (a green card or stamp in your passport).
Do consult an an experienced immigration attorney for assistance.