How does air traffic control communicate with an aircraft?
1) Why doesn't the tower use satellite throughout the flight?


The narrow answer is that the control tower is responsible for a small geographic area.  The "tower" doesn't want to talk to the craft at other points in the flight, so the need is not there.  It only wants to talk to the plane when it is close to the tower's airspace.

Other components of air traffic control that do need to talk with airplanes in flight also divide up regions of the airspace and have radio equipment situated to talk to the necessary craft.

A larger answer is that ground control and aircraft communications were under major development for decades before satellite communications were available.  The system we use today is heavily based on line-of-sight VHF communication radio over most populated areas.  Since these systems are already installed in many thousands of aircraft, changing to something different like satellite communication requires a significant benefit, someone to pay for it, and a lot of time to allow for the change.  


  2) Does a control tower only send voice services through the direct
  link? or does it send data as well?


Data is beginning to be sent via CPDLC.  You can search for CPDLC on this site and find several questions and answers about it, but it is considered only a supplement to the voice communication of ATC today.


  Aren't there security issues or power safety issues involved in ground
  to air transmissions?


Unauthorized transmissions are possible, but they are managed and haven't traditionally been a big problem.  There's another question about What prevents someone from hijacking a radio frequency mid-air?.  But much of the answer is that since ATC is monitoring the frequency, unauthorized transmissions would be heard and responded to.  I'm sure that future digital delivery will change some of these issues, but that's likely to be true whether the message is transmitted via local radio transmitters or satellite.

Not sure if that's what you meant by security issues.  I'm not familiar with power safety issues.