PETROPAVLIVKA, Ukraine—Mayor Natalya Voloshina thought things couldn't get much worse after mangled body parts and suitcases from Malaysia Airlines Flight 17 rained down on homes and buildings here in mid-July.

But she was wrong. In the days since the Boeing 777 exploded, the village of about 800 people has found itself gradually gutted by the ugliness of fighting between Ukrainian forces and pro-Russia rebels.

Residents...