The character count does include spaces. While word wrap is on, if you hit return at all when you enter your essay each return is worth two characters. So, if you put a line between paragraphs, I believe that will cost you 4 characters.
I had 5298 chars when I put my essay in Word, but I had to omit a sentence when I got it into the actual application because I had multiple spaced paragraphs. Keep that in mind when you're writing.