I think so

No, but the mirror monster from the bathroom will.

I would...

Though I haven't really looked into it if I had to take a guess I would say it is a subconscious memory of being in the womb. A warm dark place of safety and protection, cut off from the outside world. I believe it's for this same reason that most people (or at least children) will sleep in the something reminiscent of the fetal position. It symbolizes safety and comfort.

I remember seeing some cartoon with a kid sleeping under the covers, and two monsters looking at him. One of them says "Oh no he's under the covers, whatever shall we do??".... They hit him with baseball bats.

Yes they will..

https://31.media.tumblr.com/73493db6a2ddccd0d9525eca1181a813/tumblr_inline_n0mw1tHzoA1soaz9h.jpg

https://www.youtube.com/watch?v=1jhZyrILcQk

its because our fears are psychological rather than real