Oratory comes from orare, meaning "to speak, plead, or pray." And orator is one who speaks in public or pleads on behalf of someone.
Laboratory comes from laborem, originally meaning "labour, toil, distress, trouble." It's a place where someone labors.
You're exactly right about purgatory.
A curator is one who guards or oversees something. It's from the Latin curare, meaning "to care for" (though curare and care are not related).
The root of factory is facere, meaning "to do or make." Interestingly, the word fact shares the same origin, beginning with the meaning of "something done" and then progressing to "something that has happened" and then to "something known to be true."