My daugthers 2003 Olds Alero (3.4) isn't starting. Turning the key doesn't ingage the started. First time: I went in and tapped on started and couldn't get anything to happen then all of a sudden she turned the key and it started. She took her car into the shop and they replaced the battery and cleaned connectors. Worked fine for 3 weeks. Second time: Car won't start, won't turn over. Starter doesn't engage. If I cross arch the sylinoid and starter with screw driver starter engages and turns over motor fine. Plenty of Power from Battery. Questions: Doesn't this see like a sylanoid problem? Why doesn't cross arching the sylanoid and starter start the car.. is there some kind of security device stopping that from happening? Any other ideas? Fuel pump?

This sounds like there is a dead spot in your starter. The easiest way to resolve this issue is to replace your starter. Many times, if you take a lot of short-distance trips over the life of your car, the wiring inside the starter itself will wear (short trips means starting it more often, causing a lot of wear over time). Then the inner wiring of the starter wears too much, it forms a "dead spot" -- when the starter is stopped at a very specific spot i will not form the magnetic force needed to torn the starter motor. This is a common issue which can be dealt with for a time if you have to save the money to replace the starter. Tapping the starter, or arcing it with a screwdriver makes the inner workings of the starter jump a little, allowing it to move out of the dead spot and kick over as normal, but it is only a temporary fix. Unless you refurbish electronics for a living, it is very unlikely that you will be able to repair the starter on your own, and such a repair takes some time, so you are better off to replace it than try to repair the starter itself.
As long as the starter does not happen to stop in the dead spot, it will work as normal ... but in time the dead spot will spread, causing the issue to worsen. You can keep forcing it to turn with the two tricks you used until you have had the opportunity to earn the money to replace the starter (or until you have a day off to get the repairs done) .. but over time this can cause more damage to other parts of the car, so I recommend replacing the starter as soon as you can.