There were a lot of heresies after the ecumenical councils; the monophysite controversy being one of the most famous. Considering the ecumenical councils, including Council of Chalcedon, was sponsored by the Empire, did the people deemed as heretics still see the Emperor as what he was?

Additional question which piques my curiosity: when they see the Emperor as "Christ's regent", which one that they believed?

• Did they believe that the seat of the Emperor is the seat of the God's regent, meaning anyone who took the throne of the Empire is seen as holy, because it's the seat which is holy (which would make Julian the Apostate, who was famous for being a pagan, still holy)?
• Or... did they believe that it is the personal piety of the Emperor himself which signifies his position as a God's regent on earth (which would put Julian out of the list)?

If possible, I'm curious in knowing the perspective of both clergy and laymen. Thanks!