If DC electric arc furnace is used for ironmaking(a reduction process of ore), would not bottom connection be cathode and electrode be anode for optimum results? and steelmaking being an oxidation process of impurities, would it not be optimum to use bottom as anode and electrode as cathode?

• Would these two scenario prefer significant efficiency by using electrical energy not just for melting but also for refining?
• if ironmaking (carbon ) electrode is sufficiently conductive but less reactive, would it result in some portion of oxygen from reduced oxides at metallic bath to directly escape without reacting with carbon?
• It is tacitly assumed that, electric arc can sustain similar electrolysis capability of ordinary electrolytic cell. How far is this assumption correct?