For the equality $\int_{\mathbb{R}^n}\Delta_y \eta_\epsilon(x-y)u(y)=\int_{\mathbb{R}^n} \eta_\epsilon(x-y)\Delta_yu(y)$, did we perform integral by parts twice? If so, do we need any information of u on boundary? Otherwise, the integral by parts formula does not work. We can not cancel the term.
Yes you apply the integration by parts formula twice (Green's identity). The boundary terms vanish because the mollifier $\eta_\epsilon$ is by definition compactly supported.