proof of Autocorrelation property of DFT
You did not consider the conjugate property of the DFT.

The complex conjugate property of DFT states that
$\rm{DFT}\{x^{*}[n]\} = X^{*}[-k]$. 
This implies that exponent in the second line of your proof should be $-k$. Which then allows for the DFT of $x[n]$ as $X[k]$. 
A number multiplied by its conjugate give the square of its absolute value. 

And DFT is discrete in the transform domain, hence its parameter should be in brackets rather than parentheses.