You’re not entirely off, though I quibble over the term “inferior.” The Son is begotten by the Father within the Godhead whereas the Father is unbegotten. In terms of relationship, the Son is perhaps “inferior” to the Father, or obedient to the Father. However, we should also remember that a son is the image of the father, and in God this is perfectly so. In power, existence, His Being, His eternity, the Son is a perfect image of the Father, and the Son is begotten from the unbegotten Father. They differ only in relationship to each other. The Father is the God of the Old Testament (the only one revealed then), though the Church Fathers did see the Son as the actor in various theophanies of God.
In terms of Jesus on Earth, remember that He was fully human as well. He had a human soul and a human mind that existed in a hypostatic union with God the Son, and which was obedient to God. That human mind likely needed comfort, to exist temporally, a need to pray. And certainly He wished to provide an example of how we should be. The fact that Jesus had both a human and a divine will is attested to most strongly by his agony in the Garden. However, Jesus did NOT act simply like an Old Testament prophet. He made many statements that would be blasphemy in the mouth of just a prophet, and I think we as Christians take such statements too lightly if we don’t stop to think. Jesus doesn’t ask us to only believe in the Father, he asks us to believe in him. He tells us that he is the only way, that he gives the law, that he is greater than the temple, that he is the divine I AM, that those who’ve seen him have seen the Father, that he forgives sins. He sets himself up as the true torah, priest, and king. He goes beyond what the prophets would say, and this is why people accusee him of blasphemy and tried to kill him on multiple occasions. There are good works that delve into these radical and even frightening (when properly considered) statements of Jesus. I cannot do so well here.
Anyway, our Orthodox brothers refer to the “monarchy of the Father” within the Trinity. We don’t use the term as much as Catholics, but we do recognize a patriarchy of the Trinity, the familial Fatherly superiority in the relationship.
P.S. Also, question the Father on the cross? Do you mean the “My God, my God, why have you forsaken me”)? That is the opening line to Psalm 22. We should follow that psalm to its conclusion, which ends in proper vindication for the persecuted holy one and the triumph of God. We see the psalm’s truest fulfillment in Jesus. Jesus, at this moment, was also experiencing the pain of sin (that of others) for the first time, which should be considered.