Is it normal for a laptop to be still running after it's subjected to impact?

Depends on how well the laptop is made. I had one that fell down my front porch steps and still worked, and it was on as well. I am not sure about the Hard Disk or why Firefox won't launch. Try running virus scans, maybe in between the time of those problems and the time you dropped it you got a virus on it from a site you visited... I'm not sure though.