If you feel comfortable in your system you could strip it down to the bare bones (1 stick ram, cpu, gpu, hd) and see if you are able to achieve stability. Add parts one by one to find a hardware issue (if this is the case)
Also since you have an older processor have you reapplied thermal paste lately? Intel often builds fail safes into their processors so if it gets too hot it will lag itself to keep in an acceptable temp range. I thought of this because you said it shuts down on you when your playing games at night (after the computer has been on all day?) and is fine in the morning (time for processor to cool down). If your cpu is improperly cooled you can get a slew of nasty problems/crispy chips.