lack of sleep + caffeine= shakiness and anxiety?

Yes.

Caffeine causes your body to produce adrenaline, adrenaline is a stress triggered hormone. The same would release if your body were injured or experiencing some type of trauma. But this isn't occurring naturally,it's induced. Too much of it does stress the body. And in even more excessive amounts it can affect your heart.

Could be one reason but you'd be better off seeing your gp incase it's something else.

yeah it is normal, caffeine is a poison.

yes it is, go get some sleep

yes. the more caffine you have and the less sleep you get-the worse it will become.

True!!

It will make you sick. so please do not do that.