The sniper kills the old woman because she is an informant for the Free Staters. 

In addition to her being an informant, the old woman knows the location of the protagonist sniper.  The sniper watches her flag down an enemy vehicle.  She talks to the soldier inside, and she points toward where the sniper is.  At this point in the story, three people know the location of the Republican sniper.  The enemy sniper, the old woman, and the soldier in the vehicle all know where he is hiding.  In order to safely exfiltrate himself from his location, the sniper needs to kill everybody that knows where he is.  The sniper successfully shoots and kills the old woman and the soldier on the street.  Unfortunately, taking those shots exposed him to the enemy sniper, and the protagonist is wounded.  The wound prevents him from being able to effectively hold his rifle.  In order to survive, the protagonist comes up with a plan that allows him to use his sidearm to kill the enemy sniper.   

Why does the sniper kill the old woman, and what happens to him afterward?

The sniper, from The Sniper by Liam O'FLaherty, kills the old woman in the street because she is pointing out his position to the other side.  She is an informer.  Because she is putting his life in danger, he shoots and kills her.  This is war, and even cilvilians, like the old lady, are fair game.

Immediately after he kills the old woman, another sniper on a rooftop across the street knows his position and fires.  The sniper is hit.  He is shot in the arm, which renders his arm useless.