Describe some tests you might perform to prove if an unknown solid material is an element or a compound.

Always remember that a compound can be separated into simpler substances by chemical methods/reactions. While elements cannot be broken down into simpler substances by chemical reactions. You can do a flame test and spectrum analysis to determine whether a solid material is an element or a compound. Check the boiling and/or melting point, color or density. Also check the solid material’s reaction with oxygen, hydrogen, calcium, or various acids. Examine and study its physical chemistry. The element(s) that may be present may be identified by checking the absorption edges from an x-ray spectrum.

For instance, a solid white material (A) is heated in the absence of air, decomposing to form a new white material (B) and gas (C). Gas (C) has the same properties as the products when carbon is burned in an excess of oxygen (CO2). How can we determine whether A, B, and C are elements or compounds? A would be a compound since it was broken down into B and C. So how do we determine B and C?

A compound can be decomposed, for the reason that a compound is composed of 2 or more elements. On the contrary, an element cannot be decomposed.  

oxygen, selenium, tellurium, and polonium.

they're all from the same group as sulfur (group 16).

while cooking a piece of 1/2 pound beef steak, a cast iron pan (2.4kg in weight) was heated 260 celsius.