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Where does it say in the New Testatment that praying for the souls of the dead will have any impact on God's Judgement of them?

That's an excellent question and one that needs repeating. Believers in Jesus Christ make a decision, either consciously or otherwise, as to what constitutes "source material" for their faith. One of the big things that Catholics, Protestants, and Fundamentalists (since the Catholic Organization was not in existance for at least 300 years after Jesus Christ, I place the early believers and those who want to have teachings like theirs in this category) disagree upon is the idea that the Bible should be the only source of information about God.

If a believer allows other sources(aside from the bible) to impact their thinking towards the will of God, treating these documents and people as inspired, then you have to next ask where to draw the line? Just as an example, some of the Popes throughout history were pretty wicked little fellows. So were they infallible and did they speak for God? Which of their writings are divine and whose to say?

I've chosen to stick with the Bible as The Word of God.

...since the Catholic Organization was not in existance for at least 300 years after Jesus Christ..

This is not on topic really, but you might want to do some reading here. Get a good book on the Church Fathers. You will be surprised to learn, as I was, that the "Catholic Organization" was thriving early and everywhere.