Is the area under the Cantor Lebesgue function a Borel set of $\mathbb{R}^2$? If $f$ is the Cantor Lebesgue function, then is the set $\{(x,y) \in [0,1]^2 : 0\leq y \leq f(x) \}$ Borel ?

Yes: Cantor's function is continuous, therefore its hypograph is closed in $(\operatorname{domain})\times\Bbb R$. Specifically, that set of yours is $([0,1]\times[0,\infty))\cap g^{-1}(-\infty,0]$, where $g:[0,1]^2\to \Bbb R$ is the continuous function $g(x,y)=y-f(x)$.
