Must compounds formed between solids be a solid as well? My teacher gave us a long list of compounds and asked whether they are solid, liquid, or gas at room temperature and normal atmospheric pressure. I only know the more common compounds (eg. carbon dioxide, ammonia) and elements but not the rest. 

Can I assume that a compound is solid because its constituent elements are solids at normal temperatures and pressures as well? If so, does the same goes with liquids and gases? If so, what if its constituting elements are not of the same physical state?

If not, then how can I determine the answer? Or must I memorize the physical state of the compounds in order to answer the questions?
No, they don't have to. In short, you have to memorize. The physical state of a compound has nothing to do with the states of its elements.

Examples:
$$\ce{Hg~(liquid) + Br2~(liquid) -&gt; HgBr2~(solid)}$$
$$\ce{C~(solid) + S~(solid) -&gt; CS2~(liquid)}$$

Now, there are some common patterns (an ionic compound is most certainly a solid; a compound with small non-polar molecules is probably a gas), but that's another story.