I would say you have a short in the electrical system. I assume it's an Auto tranny, if so, then there maybe a short from the reverse light switch that is shorting out the ignition. Look and see if you have wiring problems under the shifter and follow the reverse light switch wires.

I am not a Volvo expert by any means, it just seems obvious that if the engine quits when put in reverse then it is probably an electrical fault. The reason it splutters when going down hill is because the tranny is turning a dead engine. An auto transmission will turn the engine when coasting at reasonable speeds.