They’re all correct.
I don’t have pencils
A store clerk might say this when he has sold all his pencils or doesn’t carry them.
I don’t have a pencil
You might say this in a classroom when you need a pencil.
I don’t have any pencil
This sounds like the answer to, “But you must have SOME kind of pencil (a broken one, a short one, etc.).” The answer means, “No, I don’t have any kind of pencil at all.”
I don’t have any pencils
This is for a situation where you probably should have several pencils, but you don’t have any at all. A store clerk might say, “I don’t have any pencils right now. I’ll have some tomorrow.” A teacher might say, “My kids need to write, but I don’t have any pencils.” My mother might have said, “We don’t have any pencils. I’ll have to get some today.” (This would have meant there were no pencils in the pencil drawer, and my mother would have needed to buy a package of them.)
There is nothing grammatically wrong with using as instead of because in formal writing. However, in most of the English-speaking world, you will sound a little strange if you use as instead of because when you are speaking.