First of all, its simple enough to get the right apparatus to check melting points that the setup SWIY suggests is just silly. Its called a thiele tube kids, you pour some oil in there, heat the sidearm and watch your capilary tube ruberbanded next to your thermometer until you see it melt
Second: an impure substance will always melt lower then any of its components, going down as low to what is known as the eutectic point. This effect is very evident in pipe solder: the alloy of the two components melts drastically lower then either of the components. You wont see anything "melt away" reliably at the correct temperature, this is NOT a distillation. Period. However, you are correct in stating that the purity of a material can be judged by its melting point/range. However, there is no definite formula SWIM
can remember....so its mostly assumption, depending on the chemical and the impurities.
You should also note that not all compounds will melt before being oxidized: caffeine
and sucrose are two commonly encountered "cuts" that will oxidize before or at thier melting points
Beyond that...decent post, and welcome ^_^