First of all, sorry for the picture – it was something that would have taken me quite a while to LaTeX up.
Below is my workings on the problem stated in the title: calculating $\mathrm{Ext}_R^i(\mathbb{Z},\mathbb{Z})$ for $R=\mathbb{Z}[x,y]/(xy)$.
From what I can see, all of the $\mathrm{Ext}$ terms should be zero, since the cohomology of $\mathrm{Hom}(F_\bullet,\mathbb{Z})$ is zero, where $F_\bullet$ is an $R$-free resolution of $\mathbb{Z}$.
But the next question in this exam paper asks to show that two specific obstruction classes generate $\mathrm{Ext}^1_R(\mathbb{Z},\mathbb{Z})$, and so this group can't be zero. Where is my mistake?
Your resolution seems fine, but when you apply $\text{Hom}_R(-,{\mathbb Z})$ to it, the differentials will vanish: $\text{Hom}_R(R,{\mathbb Z})\cong {\mathbb Z}$ as $R$-modules, and $x,y$ act trivially on ${\mathbb Z}$ by the latter's definition.