Step 1: For the hemisphere to rest completely on the top face of the cube, the diameter of the hemisphere must be at most the edge of the cube.
Step 2: Therefore the smallest possible edge is
$$a=\text{diameter}=2r=2\times 3.5=7\text{ cm}$$
Step 3: Total surface area of the solid $=$ (TSA of cube) $-$ (area of the circular base of the hemisphere) $+$ (CSA of the hemisphere).
$$\text{TSA}=6a^{2}-\pi r^{2}+2\pi r^{2}=6a^{2}+\pi r^{2}$$
Step 4: Compute the surface area of the cube.
$$6a^{2}=6\times 7\times 7=294\text{ cm}^{2}$$
Step 5: Compute $\pi r^{2}$.
$$\pi r^{2}=\frac{22}{7}\times 3.5\times 3.5=38.5\text{ cm}^{2}$$
Step 6: Add the two results.
$$\text{TSA}=294+38.5=332.5\text{ cm}^{2}$$
Hence the smallest edge of the cube is $7$ cm and the total surface area of the solid is $332.5$ cm$^{2}$.