Is there a converse of "judge not, lest you be judged?"

The two passages you quote are not converses. In fact they are strongly linked. The point about "judge not, lest your be judged" is reinforced not contradicted by "all have sinned and fallen short". The point is that 'nobody is perfect', therefore making (condemnatory) judgements about others invites those same judgements to be applied to you. If you want to be treated with mercy and understanding, you should do the same to others. You are absolutelyh right that God, who is all-knowing, should be the one to judge.

That is not to say that sins 'don't matter', or that there should be no such thing as earthly justice. Jesus pronounces condemnation on those who are wicked, and there are examples of Paul and the Apostles doing the same. Passages in Romans clearly indicate the role of secular authorities in maintaining justice.

In the case of Nazis, who are of course an extreme case, we are absolutely right to try and condemn them as war criminals and punish those left alive appropriately. However maybe we should not be quick to state with certainty that even they are necessarily eternally condemned.

When you understand what you are not to judge then it makes sense.

So this is what the Sovereign Lord says: “See, I lay a stone in Zion, a tested stone, a precious cornerstone for a sure foundation; the one who relies on it will never be stricken with panic. (Isaiah 28:16 NIV)

When the truth comes many will find it offensive (Acts 7:51). We must not judge the truth (Mark 3:29). For those that exalt themselves will be humbled (Luke 14:11).

Acquitting the guilty and condemning the innocent the Lord detests them both. (Proverbs 17:15 NIV)

We "ARE" the judges for the now (James 5:9). The Spirit of Truth indeed will come to judge regulatory.

Everyone who does evil hates the light, and will not come into the light for fear that their deeds will be exposed. (John 3:20 NIV)

Either one will accept the truth, or exalt themselves above God becoming "The Lawless one" or the "Abomination of Desolation" (2 Thes 2) So it is best to flee from the argument.

The condemnation is prepared for all:

Whoever believes in him is not condemned, but whoever does not believe stands condemned already because they have not believed in the name of God’s one and only Son. (John 3:18 NIV)

But by faith one can resist the anger and remain in the vine. (John 15)

For the stages of Grief (Denial, Anger, Bargaining, Depression, and Acceptance) Display the process that God uses. From the lack of Faith our denial leads to weeping and gnashing of teeth, and through forgiveness caused by the death of the Lord Jesus acceptance can return us to the vine.

As a quote from an old time favorite movie "Forgiveness is divine, but never pay full price for late pizza."

Here are some punishments that where dealt:

hand this man over to Satan for the destruction of the flesh, so that his spirit may be saved on the day of the Lord. (1Cr 5:5 NIV)

Among them are Hymenaeus and Alexander, whom I have handed over to Satan to be taught not to blaspheme. (1Ti 1:20 NIV)

Now a man named Ananias, together with his wife Sapphira, also sold a piece of property. With his wife’s full knowledge he kept back part of the money for himself, but brought the rest and put it at the apostles’ feet. Then Peter said, “Ananias, how is it that Satan has so filled your heart that you have lied to the Holy Spirit and have kept for yourself some of the money you received for the land? Didn’t it belong to you before it was sold? And after it was sold, wasn’t the money at your disposal? What made you think of doing such a thing? You have not lied just to human beings but to God.” When Ananias heard this, he fell down and died. And great fear seized all who heard what had happened. Then some young men came forward, wrapped up his body, and carried him out and buried him. About three hours later his wife came in, not knowing what had happened. Peter asked her, “Tell me, is this the price you and Ananias got for the land?” “Yes,” she said, “that is the price.” Peter said to her, “How could you conspire to test the Spirit of the Lord? Listen! The feet of the men who buried your husband are at the door, and they will carry you out also.” At that moment she fell down at his feet and died. Then the young men came in and, finding her dead, carried her out and buried her beside her husband. Great fear seized the whole church and all who heard about these events. (Acts 5:1-11 NIV)

All of these are deliverance of condemnation resulting from a judgement. Indeed it was the truth working through the apostles. That is why we fear the truth instead of man.