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.

Concerning Question 3: If $X \subset Y$ are 2-complexes, then the induced map $H_2(X) \to H_2(Y)$ is injective. This follows from the long exact sequence in homology. If $Y$ deformation retracts onto a 1-complex, then this implies that $H_2(X)=0$ and hence $\chi(X) \leq 1$ if $X$ is connected. I do not see how $\chi(X)=1$ could imply that $X$ is contractible. 

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