The answer is yes for the smooth enough boundary, provided $\varphi|_{t=0}=0$ and for 2) additionally $\partial_t\varphi|_{t=0}=0$. Say, for the SLP from $C^{1+\alpha/2,2+\alpha}(\bar{Q})$ should hold $u|_{t=0}=\partial_tu|_{t=0}=0$. From here the mentioned conditions on $\varphi$ at $t=0$ follow.

The answer is yes for the smooth enough boundary, provided $\varphi|_{t=0}=0$ and for 2) additionally $\partial_t\varphi|_{t=0}=0$. Say, for the SLP from $C^{1+\alpha/2,2+\alpha}(\bar{Q})$ it should hold $u|_{t=0}=\partial_tu|_{t=0}=0$. From here the mentioned conditions on $\varphi$ at $t=0$ follow.