"Why have you forsaken me?"
Him saying that, is that a lie?
I'm sure he suffered... I'm pretty sure at some point, even he lost faith. How would it be a sacrifice, if he didnt feel pain, and if he wasn't broken at some point?
Suffering can bring strength. We suffer, to learn.
You're right to say he suffered and may have started to lose faith and even doubted. Being fully human he would have gone through all the emotions we feel.
Jesus' experience in the garden of Gethsemane is a good illustration of his human nature/flesh battling with his divine nature/spirit. We can see his human nature had doubts.
Let me condense a few verses in Matthew 26:36-46 to show this:
Then Jesus came with them to a place called Gethsemane... He began to be sorrowful and deeply distressed... “My soul is exceedingly sorrowful, even to death.” ...
He went a little farther and fell on His face, and prayed, saying, “O My Father, if it is possible, let this cup pass from Me; nevertheless, not as I will, but as You will.” ...
"The spirit indeed is willing, but the flesh is weak." ...
Again, a second time, He went away and prayed, saying, “O My Father, if this cup cannot pass away from Me unless I drink it, Your will be done.” NKJV
Luke 22:42-44
“Father, if it is Your will, take this cup away from Me; nevertheless not My will, but Yours, be done.” Then an angel appeared to Him from heaven, strengthening Him. And being in agony, He prayed more earnestly. Then His sweat became like great drops of blood falling down to the ground. NKJV
It's evident that Jesus' human will did not want to bear the cross, and he even prayed to the Father not to go through it, but nevertheless he said chose "not my will be done but Your will be done." He was completely surrendered and submitted to God's will.
It translates to: "My god, my god, why have you forsaken me?"
Implying he was not God.
That verse can be explained by a long theological response, but I'll keep it short and simple. At that very point Jesus became sin on the cross. The divine nature was dissociated from Jesus, and God the Father could not associate with sin. Thus Jesus felt forsaken because the Father turned his face from him - that is he turned his face from sin.
Throughout all the Gospels you will see many people call Jesus "Lord" yet he never rebukes them.
The incredible power and glory of God was revealed when Jesus rose from death.
Matthew 28:17-18
When they saw Him, they worshiped Him; but some doubted.
And Jesus came and spoke to them, saying, “All authority has been given to Me in heaven and on earth." NKJV
Mark 15:37-39
Jesus cried out with a loud voice, and breathed His last.
Then the veil of the temple was torn in two from top to bottom. So when the centurion, who stood opposite Him, saw that He cried out like this and breathed His last, he said, “Truly this Man was the Son of God!” NJKV
To give you background on the significant of the veil in the temple being torn after Jesus died and the earth quaked; behind the veil in Jewish temple was God's earthly dwelling where no one was allowed to enter into except for the high priest once a year to make atonement for the sins of Israel. The veil symbolized the separation of God from his people because of sin. Above all, the tearing of the veil at Jesus' death showed that his atonement was a momentous event that unleashed God's presence to be open to all people, for all time, both Jew and non-Jew. We can now freely enter into God's presence and God's forgiveness because of Jesus and the tearing of the veil.
The Gospels, in their entirety, show that Jesus is God incarnate, I will leave you with this:
John 20:26-29
Now Thomas, called the Twin, one of the twelve, was not with them when Jesus came. The other disciples therefore said to him, “We have seen the Lord.”
So he said to them, “Unless I see in His hands the print of the nails, and put my finger into the print of the nails, and put my hand into His side, I will not believe.”
And after eight days His disciples were again inside, and Thomas with them. Jesus came, the doors being shut, and stood in the midst, and said, “Peace to you!” Then He said to Thomas, “Reach your finger here, and look at My hands; and reach your hand here, and put it into My side. Do not be unbelieving, but believing.”
And Thomas answered and said to Him, “My Lord and my God!”
Jesus said to him, “Thomas, because you have seen Me, you have believed. Blessed are those who have not seen and yet have believed.” NJKV