The victim, who records claim was the woman's boyfriend or husband, was working under the car before he was fatally run over and dragged, police said.
HPD said the 29-year-old driver stopped after the crash and waited for police, who determined she was intoxicated and arrested her.
The apartment where 2-year-old Egypt was fatally injured was quiet on Thursday, but the night before had a flurry of activity, with reports of several men walking around with guns.
The sheriff's office says one person was flown to the hospital in critical condition. The suspect fled the scene in a dark-colored sedan.
Following the crash, police say the driver, who was questioned and did not appear to be impaired, was released after.
Investigators said the person who fired the shot may not face charges at this time.