Judging by the various symptoms and the ongoing misbehaviour, I would be inclined to try the "minimal spec" startup - just power supply, motherboard, memory, and boot hard drive.
However, from the problem description I would point a stern accusatory finger at the power supply. Spiking voltages, or over-volts, can cause all kinds of hiccups. A good-quality power supply, rated at more watts than the PC uses, may resolve the problem.