This answer concerns Question 1. Take a free group on $2$ generators and $k$ very long relations at random. Then, with high probability, the representation satisfies a small cancellation condition and the associated 2-complex $X$ is aspherical. Moreover, if this happens you also have $\chi(X) = k-1$.

A negative answer to Question 2 would also be a negative answer to [Whitehead's Asphericity Conjecture][1]. I believe that a positive answer is not easy to obtain either.


  [1]: http://en.wikipedia.org/wiki/Whitehead_conjecture