The Council of Chalcedon of A.D. 451 declares that Jesus Christ is One Person who has Two Natures:
Our Lord is truly God and truly man, of a reasonable soul and body, consubstantial with the Father according to the Godhead, and consubstantial with us according to the manhood; in all things like unto us without sin; begotten before all ages of the Father according to the Godhead, and in these latter days for us and for our salvation born of the Virgin Mary, the mother of God according to the manhood; one and the same Christ, Son, Lord, only begotten, to be acknowledged in two natures, inconfusedly, unchangeably, indivisibly, inseparably, the distinction of natures being by no means taken away by the union, but rather the property of each nature being preserved and concurring in one person and one subsistence; parted or divided into two persons but one and the same Son, and only begotten, God, the Word, the Lord Jesus Christ. Source: God the Father, God the Son, page 281 by Dr. Martyn Lloyd-Jones, Westminster Chapel, London, England.
One person, two natures, the two natures unmixed, joined but not mixed, not fused, not intermingled, remaining separate, God and man. In Reformed doctrine, the divine nature and the human nature are united in the person of Christ.
The Council of Chalcedon anathematized those who taught that Christ had only a single, divine nature and those who taught a “mixture” of His two natures. The “Chalcedonian Definition” affirms that Christ is “the same perfect in Godhead and also perfect in manhood; truly God and truly man.” He is “consubstantial [homoousios] with the Father according to the Godhead, and consubstantial with us according to the Manhood.”
By affirming that Jesus Christ is one Person who is both divine and human, the Council of Chalcedon made it easier to identify error. The Chalcedonian Definition affirms the truth that Jesus Christ is fully divine and, at the same time, fully human. He is both the Son of God (1 John 5:10) and the Son of Man (Mark 14:21). Jesus, the Word incarnate, assumed perfect humanity in order to save fallen humanity. He could not have saved us unless he was fully God and fully man.
Jesus said his soul was exceedingly sorrowful unto death (Mark 14:34). His suffering involved the body and the soul, so He had to be a man. In order that His sacrifice might have infinite value, He had to be God as well as man.
Paul says had they known it, they would not have crucified “the Lord of glory” (1 Corinthians 2:8). You cannot crucify God (who is Spirit) but the body of Jesus could be crucified. Yet Paul does not say that His body was crucified. Paul says they crucified “the Lord of glory”. In other words, what happens in the one nature or the other is ascribed to the one person.
When Thomas fell down and declared that the risen and resurrected Christ was his Lord and his God, he was acknowledging him as both Lord and God. Be assured that Thomas was not worshipping two Gods. He acknowledged the reality of Jesus’ divine nature whilst also holding to the monotheistic doctrine that there is only One God.
You ask if Thomas “believed God could have and did die”. If by “death” you mean “a cessation of existence,” then, no, God did not die. Neither the Father nor the Son nor the Holy Spirit will ever cease to exist. The Son left the body He temporarily inhabited on Earth, but His divine nature did not die, nor could it. Thomas was neither a Unitarian nor a follower of Arianism.
To conclude: The Logos, the Son of God, the Christ, became man that the children of men might become children of God. Thomas finally saw the light (spiritually speaking) when he fell down on his knees and cried out in reverent worship, “My Lord and my God.”