What does the monitor do? Would Ender have taken the same action to defend himself in chapter one if he'd still had a monitor?

This device had allowed the authorities to view the world as Ender did. From Ender's thoughts we learn that the monitor has made him an outcast. When Ender makes the decision defend himself and beat Stilson.... and then beat him some more, he does so in order to minimize or eliminate the possibility he will ever be bullied again.

As for the second part of your question, I believe that the absence of Ender's monitor encouraged the other kids and their bullying more than it influenced Ender's reaction. The monitor kept Ender safe. It recorded everything that was done and said to him. When the monitor was removed, Ender lost that protection. Stilson, and others like him, now had free reign to do and say whatever they wished to Ender without fear. Thus, Ender fought, and fought to win, in order that it would be his last fight. Unfortunately, his battles were just beginning.