One thing you didn't mention is whether you tried a good glow plug. We'd suggest an O.S. #8. Usually, when an engine quits when you remove the glow plug battery, you have a bad plug.

Starting backwards almost always means that it's flooded or too rich.

Sudden cutoffs when running usually mean that something's awry with the fuel system or carb adjustment.

Model two-stroke engines are very simple. there's almost nothing internal that could cause your difficulties unless the engine's had some kind of accident, such as a prior overheat or foreign object ingestion. They could cause a problem with the fit of the piston/ring/liner assembly, which could cause the engine to cut off once it's warmed up. Wouldn't cause the backfiring or backward starting, though.

To solve any difficulties, we'd need to see the engine.