If $u(x,t)$ satisfies the heat equation then $\eta_{\epsilon}*u$ also satisfies it, with $\eta(x)=e^{\frac{1}{|x|^2-1}}$ for $|x|<1$ and $0$ else and $\eta_{\epsilon}(x):=\frac1{\epsilon^{n}}\eta(\frac{x}{\epsilon})$
It is not mentioned which heat equation (homegeneous or not). I think, one has to get then;
$\partial_t(\eta_{\epsilon}*u)=(\eta_{\epsilon}*\partial_t u)$
$\Delta_{xx}(\eta_{\epsilon}*u)=(\eta_{\epsilon}*\Delta_{xx}u)$
then the difference should be zero, but which property is used to justify the differentiaion under the integral, that the support of $\eta$ is bounded or that both functions are twice differentiable
You are definitely using that $u$ is twice differentiable in $x$ and differentiable in $t$. You also need to show/assume that $\eta_{\varepsilon}\in L^1(\mathbb{R})$.