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. 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. Indeed, any knot complement deformation retracts onto a 2-complex, which has the property that adding one 2-cell makes in contractible. It has Euler characteristic 1, but is not contractible.
