An obvious answer is to why the presence of carbon, per a source Chapter 4: The Role of Carbon in Metallurgy on the historical extraction of iron from its ores, for example, to quote:
Carbon plays a crucial role in the chemical preparation of metals, particularly iron. ... This chapter discusses carbon as an intermediary product used in very large quantities to reduce and insulate metals and semi‐conductors from their natural oxides.
In a few words, carbon is a reducing agent, an insulator and a good oxygen scavenger.
Also, given the electrolysis nature of the Aluminum process, it is a cheap and very noble cathode. The employment of other noble metal cathodes would be more expensive and likely add impurities.