Hydrostatic pressure and sump pumps
why won't a sump pump help? If water is trying to get into the
  basement wouldn't pumping it out alleviate the problem?


A sump pump is designed to be located in a low area, usually a basin or pit, and is for removing water that is actually in the room.

The hydrostatic pressure that you are talking about is the pressure of the entire water table in the earth surrounding your home. There is no way you could effectively lower the entire water table (and subsequently reduce the hydrostatic pressure on your basement walls and floor) with a single sump pump.


  ...will installing a sump pump cause the water to overflow the hole
  and flood the basement...?


Yes, if I am understanding what you are proposing. If you cut a hole through your basement floor and dig a pit, the surrounding water that is exerting hydrostatic pressure on your basement walls and floor would come inside, it would be a bad day for you.