What is the crystal structure of Calcium Acetate?
Too long for a comment, hence an answer:


What you are asking about is not crystal crystal structure, it's called crystal morphology. Crystal structure is a model describing atomic positions and overall symmetry within a unit cell, whereas crystal morphology is the appearance of the crystal on a macro-scale. And yes, crystal morphology is also affected by the conditions crystals are growing under (concentration and evaporation rate, in your case), so that the same substance can have crystals of different shape.
The formula for calcium acetate and hence the equation is wrong, the correct one would be $(\ce{OAc-} = \ce{CH3COO-})$
$$\ce{CaCO3 + 2HOAc → Ca(OAc)2 + H2O + CO2}$$
and, since your reaction's stoichiometry suggested 1:1 ratio between calcium carbonate and acetic acid, I suspect you didn't use enough acid leaving $\ce{CO3^2-}$ in the solution. Then, as the primary product – calcium acetate $\ce{Ca(OAc)2}$ – has crystallized (white edgy crystals on the bottom), the remaining calcium carbonate $\ce{CaCO3}$ started to precipitate or co-crystallize on the walls of the beaker (what you call a "styrofoam look-a-like" phase), resembling how calcite grows in nature (see this and other images of natural calcite on MineralMiner's FTP server):

Also note that there is no such thing as $\ce{CO3}$. It's either carbonate $\ce{CO3^2-}$ (solution) or carbon dioxide $\ce{CO2}$ (gas).
Regarding the the impurities from the chalk, usually there are parts of shells of marine animals and silicates (can be eliminated by filtering), also minor amounts of $\ce{MgCO3}$. However, if you took a chalk bar from the classroom, this is a different story as it's often nearly 50% $\ce{CaSO4}$ (I hope you didn't do that).