I used Vaseline petroleum jelly on a deep scratch on one of my monitors before. It didn't remove it completely but its no longer that visible.

There is no real way.

You could try a diamond lapping paste. I have removed scratches from my watch and windscreen before, the plastic screen should submit to the polish much more easily than glass and sapphire. You might have to make it worse before better with some emery paper though so you don't just replace the scratch with a mirror shiny stripe though.

I wouldn't bother personally.