Not that it would matter in a simple Pong game, but in a more "to-scale" sort of scenario, does maintaining multiple VBO's impact performance in any significant way?
Kinda. You want to keep the number of GL calls to a minimum. In the most ideal case, you'd call glDrawElements just once per frame. You can get away with a lot on modern hardware though, so unless you're doing something that really pushes the limits of the machine, you should be able to break it up however is most convenient for you.
In response to your general questions about managing paddle position and drawing, there are a lot of different ways it could be done. Since you're going the VBO route from the start, the way I'd do it would be to keep paddle position in a model object that knows nothing about drawing, in whatever coordinate system is convenient. When you want to draw it, have a view object read the model's position, then construct its vertices based on that and buffer them into a VBO with GL_STREAM_DRAW. You can set up your view matrix however you want; it's probably most convenient to match it to your paddle model's coordinate system, but if you wanted your vertices in the range of 0..1, some simple division can transform one coordinate space to the other.
I'd recommend not putting too much into your vertex shader until you're sure you need to. Better to learn the simpler and more direct ways of doing things first to have a good foundation, then use fancier techniques later if you're in a situation where they become necessary.