I think you could use 3 papers or 3 sheets of paper. I'm guessing the reason for the variation is the age of the writing. In olden day ( :-) ) there were no photocopiers so you would not just hit "Print". So when he was writing 3 papers would mean three individual papers. I even looked out some old "exam papers" and, before the 1950s they would try and print everything on one side of a piece of paper even at University level. So, I guess now-a-days we would write 3 sheets of paper. But 3 papers is ok too, just a little uncertain to our modern ears.

When referring to paper as a material, it isn't pluralized. Yes, it is a noun but the word itself doesn't refer to a specific object. You generally wouldn't say "three steels", you would say "three bars of steel". Likewise, you would say "three sheets of paper", so you are right. However, paper has another definition. This one can be pluralized. Some documentation (such as government identification, medical records etc) are sometimes referred to as "papers". Exams are also sometimes referred to as papers. In the passage you described, they are using "paper" by this definition.