If I've understood your first question correctly then you can take $g$ and $h$ to be the sum of monomials corresponding to the vertices of $Q$ and $R$ respectively. Then $\tilde{f}=gh$ does the job.
For your follow up question, the answer is negative. Consider $S_g=\{(0,0),(1,0)\}$ and $S_h=\{(0,0),(0,1),(0,2)\}$ and $S_f=\{(0,0),(1,0),(0,2),(1,2)\}$. Since $g=a+bx$ and $h=c+dy+ey^2$ for non-zero $a,b,c,d,e$ the coefficient of $y$ in $gh$ is $ad\neq 0$ but $(0,1)\notin S_f$.