I'm not sure of the deeper explanation, but in Italian, when asking someone how they are, the correct question is "Come stai?" and the appropriate response is "Sto bene/male/etc." Stare also used to express ideas like "I am learning" ("Sto imperando") or sometimes when telling someone how to act, eg. "Stai tranquillo" (Stay calm/quiet). I tend to think of it as one of those idiosyncrasies where you just have to memorize which expressions take stare and which take essere.
Based on just your description, it sounds like "stare" is used more to describe a person's state (is eating, is learning, is well) vs "essere" used more to describe what a person/thing actually is (is a woman, is a doctor, is a monkey). Does that seem to extend beyond your examples, or is it really going to boil down to memorizing still?
Essere is basically just used to mean "to be," "to be in the state of having (done something)" or "to have passed" - depending on context. Stare, on the other hand can mean to be, stay, remain, keep, stick, live, be up to, etc. depending on context. There is overlap, but stare can be used many places essere cannot. If you think of stare as roughly "to stay" and essere as "to be" (being derived from the Latin verb for to be) then it makes more sense, but they do have somewhat broader uses.