Why would a piece of sheet music use the key of C# rather than the key of Db? Or the key of G# rather than Ab, etc.? Does this happen a lot? Thanks

On piano I don't think it necessarily matter much with the enharmonic keys, but in other instruments or voice, or example, singing a sharp is much higher than singing a flat.

And is there a key of G#? :P

edited to add: Unless you are talking about naming a note G# vs. A flat. That all depends on what key the piece is in and what is going on in the phrase. If something is moving up stepwise, for instance, you wouldn't want it to be G-G# B-flat, becuase then it doesn't look like it is moving stepwise.

Well notice first that C# or Db are really the same thing. Now go look at a circle of fifths and you will notice there are three duplications near the bottom of the circle.
http://numbera.com/musictheory/theory/images/fifths2.gif
C# / Db and then B and Cb and F# and Gb. Now why use C# or Db. Db has 5 flats and C# has 7 sharps.

With piano probably easier to use C# and play the black keys, however on other instruments Db would be the choice.

And yes each (full)scale should have all the letters.
No duplication of letters, i.e. Cb, C, C#.
All sharps or all flats, i.e. don't mix the two.
But in our World that gets broken all the time, i.e. the blues scale 1, b3, 4, 4#, 5, b7 or Dominant Lydian 1, 2, 3, 4#, 5, 6 b7.

If it was easy every one would be doing it.