Notting your statement that water is running in like someone turned on a faucett... Your sump pump is simply doling it's job. As the water rises it triggers the float switch / sensor and activates the pump. The pump has enough capacity to empty the catch basin to the point the float / sensor switch shut the unit off. The pump is thermally protected and will stop if there really is a problem. You home electrical circuit more than likely has either circuit breakers or fuses. From what I am reading none of those have tripped or blown. Once again the sump pump is simply working normally based out your description of the the amount of water coming in. Also not that if it did not run you would be faced with a flooding situation. Relax and enjoy your day :)

By the sounds of it, you are not in a very snowy climate in the northern hemisphere, or you are in a heat wave. the sump is used for getting rid of ground water near the footing of the walls so the soil doesn't become too loose (or wash away) and your house does not sink. right now, it sounds as if the sump pump is taking the water from the ground, pumping it back on your lawn, it soaks back into the ground in to the sump, and continues over and over. see where it's unloading. the outlet tubing should let out at least 10 feet from the house. if the hose is broken or fell off near the wall of your house, the expelled water is soaking back into the ground and right back into the sump. there may also be an underground water issue. if your water table is unusually high or there is a water main break, you'll have the same problem.