Under the rotor there's a round spongey piece of something. What is it called, what's it for, and is it supposed to be oiled???

 

if you mean inside the spindle, then that's the sponge. if it's missing you can take any spongy material that keeps oil and put it in there. Yes it's there to carry oil, but it is not supposed to be drowned. 2 drops are plenty.