You first see him outside of Prof. Elm's lab, then you see him in Cherrygrove, from there on its random
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They don't, he is your rival throughout the whole game and if he wasn't present you would not have a rival.
go to the battle tower. Across The Sea To The "Great Marsh".
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.
You get to double battle Lance and Clair in the Dragon's Den after you have defeated your Rival in Mt. Moon.
you can name your rival's name whatever you want.