No, the rival is Silver. Red is the trainer you meet at the very end of the game, after beating the Elite 4 and Kanto. And both have full teams. I don't think Silver even has a Pika...
I'm not sure. All I remember was there was this one non-named trainer out in the wild somewhere with a reidiculously overpowered Chu who easily defeated my entire team every time. I know it wasn't the rival, though...