"In Doubles, If you have a Ground-type with Lighting Rod say Rhyperior, and your partner uses Discharge is the entire move drawn or will it still hit the opponent's pokemon?"
It would not draw the entire move. I know this from experience with Storm Drain Pokemon and surf. If I have a Zapdos and Rhyperior out, and use Discharge, then it will hit both opponents. The ability only activates for the opponent's moves.