Let's say for example I have 3 parts in inventory, numbered A, B, and C respectively. Parts B and C are made from part A. When part B or C are sold, I want them to be removed from part A inventory. Is this possible without having to "build assemblies"?

No, but the Adagio Bill of Material add-on can automate the process - see http://www.softrak.com/products/adagio.php?prod=Adagio%20BoM