BIRMINGHAM, Ala.—Relentless winds and rain pummeled large swaths of the South on Tuesday, causing tornadoes, sparking a flash flood emergency in Alabama and damaging homes from Texas to Virginia. The storms prompted boat rescues, toppled trees and power lines and raised the threat of flash floods elsewhere in the region. The National Weather Service issued the flash flood emergency for the Birmingham, Alabama, area at the start of rush hour, warning that torrential rains—as much as 5 inches in some areas—had already fallen and another 2 inches were possible before the storm system continued eastward. Jefferson County Emergency Management officials in the Birmingham area urged residents to stay off the roads because so many were flooded. In the Birmingham suburb of Homewood, residents huddled on the second-floor balcony of an apartment complex that became flooded. Fire department rescuers in a small boat paddled through the parking lot past submerged cars, …