#1
The E on this cheap guitar I got from a friend always pops out of the nut. I tried filing it down, but the file I used was flat, and not thin enough, so it just filed the side, making it easier for it to pop out.... what do I use to file it down deeper?
#2
to use - file nut but if you don't have one, maybe a razor plde. I have also read of people using the e-string and wrapping it around their fingers like dental floss and sawing back and forth. Who knows if this works.

However, if you deepen the slot, you may lower your action and get fret buzz. You may want to just get a new nut.