There is water in my basement and the outdoor sump pump failed. What am I doing wrong?
For the case of the power outages, there's basically two options -- 


UPS or other battery backup for the sump pump.  (which only helps for as long as the batteries hold out)
A water powered sump pump


As for the flash flood issues -- you might be able to either regrade the area to change the catchment area that drains towards the door, and possibly add a lean-to roof to reduce the amount of rain into the area, or increase the size of the pump to deal with the increased flow.

The water powered sump could also help in the case of the flash floods, as you'll have both the regular pump and the backup pump so long as the power's still on (and you route them separately, so they're not sharing an outflow route.)