SPOILER WARNING
After he helps you beat Clair and lance in a double battle, but the Pokemon is starting to get happy, so he get's to keep it.
Chat with our AI personalities
You actually get to name your rival. So, your rival is whoever you want him to be. The rival is just a strange character that happens to be eying Professor Elm's Pokemon.
Continue with the game
You can't find professor elm.
Go show Professor Elm...Obviously
No, Profersor Elm appears in the Johto region.