Is it common for people in academia to experience prolonged lack of sleep and how to deal with it?
There are three basic approaches to handling this:


Paul Erdos' approach.  Pros: it worked for Erdos, after a fashion.  Cons: you are not Erdos.
Crack, maybe have a psychotic episode, and leave academia.  Pros: it's the easiest course, and the one you're on now.  Cons: everything else about it.
Recognize that your career is not a sprint, but a marathon, and start triaging less important work in favor of sleep.  Working sleep-deprived is exactly as smart as working drunk, so you are already cutting your productivity, just in an uncontrolled manner.  Pros: Survival, possibly having a career.  Cons: you will have to say "no" to people.


The truth is, in academia there is an endless amount of work that you could be doing: you will never be done.  The only choice you get to make is whether you will choose the things that you do not do, or whether you will make self-destructive choices that mean those choices get made for you arbitrarily by the fracture points of your body and mind.