A few points. God is not amenable to ration or logic - if he were the whole thing would be done and dusted. Since man was made in the image of God, then it must be safe to believe he shares emotions. If in the beginning God was floating about in isolation, he would have wanted love, so he created a freewill mankind with the ability to love of freewill, but also, to reject or hate with freewill. There is no point in love not freely given.
Sin is not "doing bad things", sin is a state of separation from goodness, a failure to be perfect.
The words on the cross - "why have you forsaken me?" - they may have been exactly true. A complete absence of God, a total loss of anything good, then the suffering/bearing of all the ills of the world ever, past present and future is mind-boggling. No wonder Jesus asked in Gethsemane for the cup to be taken from him!
Of course he died on the cross. The Romans knew how to crucify someone. But, imagine, he was up all night, flogged (which in itself could be fatal), nailed up for at least 6 hours, a spear stuck in his side, does anybody really believe that a short rest in a tomb being patched up with a few ointments could produce an individual who would just two days later re-inspire the depressed disciples, in some cases to martyrdom. No man will die for something he knows to be a deception.
It will never be proved and equally important it will never be disproved.