You don't have to earn the white belt. Everyone starts off as a whitebelt. If your instructor uses stripes then usually you must get four stripes before you get the new color belt (ie you would have 4 stripes on your white belt before you get your blue).
The white belt comes with the gi and you can wear it on day one.
Some instructors use stripes to represent progress within the belt level. Usually, there are four stripes available, if they are used. Some will test, some will award based on performance, some will not use stripes at all.
Everyone starts off with the white belt at your school. As you progress, stripes will be added indicating that you are moving toward the next belt rank. You won't "test" for the stripes; William will present them to you when he feels you've progressed. As far as moving to the next belt color, I know that William has some form of testing procedure; of course, I've been stuck on two stripes on my white belt for quite awhile, so I wouldn't know that part yet. ;-)
Randy: Congratulations on the blue belt. I know that you worked hard and that it was well-deserved.
Texas Thai: Lately, I just have been working with William one-on-one. I got a little banged up (I'm over 40, and these old bones take a little longer to recover) when I was doing the classes, but I hope to get back into the class mode one of these days. So, for now, I stay involved with BJJ, but at a slightly slower speed. :-)