If you plugged the basis into the equation naively (e.g. directly), you'd get unsymmetric matrices, so you first integrate the equation per partes (or the green theorem, as you use in the paper).
We do not care so much about symmetry (we use UMFPACK, right?), but about the necessary smoothness of the function spaces. Laplacian requires C^2 - now you cannot use P1 (linear) elements for that! This is why per-partes and the integral formulation is done.
Yes, that is the second reason for that.
Mathematicians usually say that the weak solution contains more solutions than the original equation, but obviously this is not true.