Sudden shutdowns are very likely to be caused by hardware issues, meaning either bad operating conditions (thermal & electrical) or bad components.
Keep in mind that while your idea of a short interruption in power is reasonable, it would not be consistently perceived by looking at keyboard and chassis LEDs. Acceptable power interruptions are in the order of a quarter of a power cycle (16ms/4 = 4miliseconds). Anything higher than that may cause a failure, and that becomes very likely to happen in interruptions that persist for 20ms or more (research CBEMA curve for more info on this). This means a power interruption could be fast enough to go unnoticed by you and still cause your computer to restart.
On the other hand, power distribution companies seldom cause this type of short interruption, which leads to the most likely cause of a power interruption being your power supply. Short power disturbances would affect other household appliances too, which you should be able to notice.
As for all the plugs going into the same socket, it should not be a problem, unless they are not securely pluged. Usual power outlets are designed for a minimum of 10A, and assuming a 127V voltage level, you'd have a safe 1270VA to use, which is more than enough. 220V would mean even more power available.
Sadly, other than a bad power supply, your second most likely cause of failure would be the motherboard's power stage.
It is not an easy issue to diagnose. I would run a stress test on the computer and monitor the power and temperature parameters. Something like prime95 would do. Keep an eye for significant voltage drops once the test starts.