*Why Einstein introduced a minus sign in the definition of the second Christoffel symbol $\Gamma^\sigma_{\mu\nu}$:*    
He writes just below equation (45) in Ref. 1:

<IMG SRC="https://i.sstatic.net/lsetA.png" WIDTH="400"/>

So he wanted to identify the $\Gamma^\sigma_{\mu\nu}$ with the components of the gravitational field, and for that identification the minus sign is needed.

  [1]: https://i.sstatic.net/lsetA.png