This is a confusing question that seems to be asking many things. Try to ask a single clear question.

The answer to the question (which I'm not sure you actually asked) "was the diatonic scale designed to make pianos easier to play" is clearly "no" because the diatonic scale precedes the invention of the piano by some thousands of years. 

Remember, for the vast majority of the history of music, it was not played on keyboard instruments. It was played on wind or string instruments. If you want to see instruments on which the chromatic scale is clearly laid out, see the neck of any guitar, ukulele, or other fretted stringed instrument. 

The answer to the question (again which was unclear) "why is C sharp enharmonic with D flat" is because it is highly convenient to do so. As other answers have noted, the fundamental relationships in music are ratios of vibrations that are 2 : 1 or 3 : 2.  But it is impossible to make any combination of 3:2 ratios that works out to a 2:1 ratio! What we do then is we choose twelve notes that are each in a ratio to each other of the twelfth root of two; that number can be raised to an integer power that gives a result very close to 3:2. I wrote a series of articles about this ten years ago; see http://blogs.msdn.com/b/ericlippert/archive/tags/music/ and start from the bottom.

The answer to your question "could we have a black key between every white key on the piano?" is yes, and this arrangement would have several nice properties including making it trivial to transpose on a piano. The traditional piano keyboard arrangement makes it difficult for even experienced pianists to play a piece known in one key in a different key, say, to accommodate the range of a particular singer. The WIkipedia article on isomorphic keyboards may be of interest to you.