Is the ferromagnetism of iron understood completely? In Feynman's lecture notes, he said that it is not (at his time).
How is the situation today?
Can first-principle calculation accounts the ferromagnetism of iron quantitatively now?
 A: It depends on what you mean by completely.
Ab initio calculations of ferromagnetism are routinely done for a wide range of systems. Googling for ab initio calculation of ferromagnetism or some similar terms will find you lots and lots of examples. So in this sense the answer to your question is yes.
However in many materials whether the system is ferromagnetic or not depends on a delicate balance between the electron magnetic interactions and the exchange interaction, and when the system is finely balanced it would require an extremely accurate calculation to get the correct answer. For example iron (which is the archetypical ferromagnet) exists in three slightly different crystal structures called ferritic, martensitic and austenitic. The ferritic and martensitic are ferromagnetic but the austenitic form is not. So the same element can switch between a ferromagnet and paramagnet just by changing it's crystal structure. I'm not sure whether the theoreticians can honestly say they can predict the correct behaviour for these borderline cases.
