electrical problem
I may go out on a limb and guess its alternator related. I've seen some failing alternators cause some really weird things in the past. Sometimes the vehicle can seem downright possessed. I could be wrong though. Just a guess.
If it is dying while the engine is running the alternator is probably bad. You can take it to an autoparts store and most will test the charging system for free. If you have a meter, connect it to the battery with the engine off. It should be around 12V. when cranking, if voltage drops below around 9V the battery is weak. After the engine starts, voltage should go up to about 13.8V, if not then the alternator is bad.



