Imperial cult was the norm in pagan Roman Empire, but when it's Christianized of course the emperor can no longer be considered some kind of god. Instead, according to Judith Herrin and Averil Cameron, Roman Emperor became God's viceroy on earth. Here is a short passage from their books:
I.
Eusobios, bishop of Caesarea, ... developed the notion of human viceroy dispensing divine justice on earth in God's name with Constantine I in mind.
II.
Official art promoted the ideas of hierarchy with emperor at the peak on earth and reflecting the hierarchy of the divine, and emperor, patriarch, higher clergy, and monastic leaders alike all engaged in a performance of Orthodoxy
When the Christians speak of God's viceroy, which one they are actually referring to: the personhood of the emperor (the emperor himself) or the seat of caesar? Did they still regard "heretic" Arian emperor like Valens as God's viceroy?
By "the seat of Caesar," do you mean the actual position of emperor? The idea of the emperor as God's Caesaropapist vice-regent on Earth, at least according to all the professors I've heard on the topic, doesn't seem to accurately reflect the situation at any point in Byzantium (though imperial rhetoric may certainly suggest otherwise).
There's no doubt that orthodoxy was certainly a desirable trait in an emperor, but its absence wasn't necessarily grounds for being deposed. Iconoclast emperors are regularly vilified by subsequent historians for precisely this reason - these historians, with the benefit of hindsight, can clearly see how wrong their Iconoclasm was, but they were still emperors. Even in earlier centuries emperors tried to promote "official" stances on heresies like Monophysitism or Iconoclasm through various decrees (e.g. Zeno's Henotikon, Justinian's doctrine of aphthartodocetism, Heraclius' Ekthesis), but these mostly seem not to have worked. The possibly suspect content of these decrees notwithstanding, both of these emperors continued to rule after their proclamation, and the heresies certainly persisted long after these emperors died.