DES and using the S-Box 1
Your XOR equations are wrong.
You wrote:

    1110
XOR 0000
  = 0001 


But it's not how the XOR works. 1 xor 0 = 1, and in your example it somehow turns into 0 wich is an error. Reference the xor table: https://en.wikipedia.org/wiki/Exclusive_or

So, the correct answer for S1(x1) XOR S1(x2) is:

    1110
XOR 0000
  = 1110


and for x1 xor x2:

    0000000
XOR 0000001
  = 0000001