Dividing different units of measurement? 1) $$\frac{35 \; \text{cm}^2}{0.62 \; \text{cm}} = 56.4516$$

For the units in the answer, do you put $56.4516$ cm$^2$ or just $56.4516$ cm?

1) $$\frac{0.075 \; \text{g}}{0.0003 \; \text{cm}^3} = 250$$
Same here, do you put $250$ g, $250$ cm$^3,$ or $250$ g/cm$^3$?
What does $g/cm^3$ mean? It means you have divided a mass in grams by a volume in $cm^3$ (the slash character"/" is really a division). So yes, your second value is 250 $g/cm^3$.

In the first case, you can simplify $cm^2/cm$, just like in any equation, to $cm$. So your value is 56.4516 cm.