Did Jesus suffer on the Cross in all possible senses? I have read that it's more or less accepted that Jesus did suffer on the cross. 
What I do not understand is whether or not He suffered all possible sufferings.

Imagine a war hero captured and tortured by the enemies. The war hero however does believe that his country will win the war and thus he does not experience all the possible sufferings for he does not experience a suffering of a man whose country has been defeated in the war.

However if Jesus does not experience a particular suffering then how Jesus can save a man who suffer such sufferings for then Jesus will not be knowing the man.

So it seems Jesus did suffer all the possible sufferings on the cross -- but then he must have suffered a suffering of someone who has discovered or wrongly felt that Christianity is false. How can that be, for Jesus surely could not believe that Christianity is false?
You're thinking too hard, and not quite straight.

Obviously Jesus didn't not experience everything. Your example is pretty convoluted but lets do an easier one. As a sinless man, there is at least one emotion Jesus didn't experience: a guilty conscience. God "made him to be sin who knew no sin", so he experienced in some fashion what it was like to be guilty, but clearly there must be a dimension in which he did not experience it the way we do.

But that's to miss the point.

The point is He suffered FAR BEYOND anything we do. He bore the weight of God's wrath against sin in a way that a Christian never will. He experienced far more senses of suffering than we do. There is left to us no sense in which he would be unable to understand what we experience.

And yet I still think that whole discussion is missing the point.

It sounds to me like you're confused about what we are saved from. Jesus doesn't save us from ourselves, our our self inflicted miseries or general sufferings. His primary work was to save us from the wrath of God. Until you understand that this is the central problem which he came to solve, none of the other stuff will quite make sense. When you get that out of the way, in having the broken fellowship between us and God restored through Christ we find that our other woes begin to heal as well -- the new creation we are is able to deal with suffering in a way that it never could before -- and as all of creation will one day be restored, the sufferings of a fallen world will likewise be wiped away with it.