Yes, you still have to discard, and no, you still cannot use Dragon Rush on the next turn (unless Garchomp is switched out).
First off, Unown G protects the defending Pokémon from effects of an attack - excluding damage - done by the opponent's attacking Pokémon. Meaning if you attack a Pokémon with Unown G, and your attack places an effect on your own Pokémon, the fact that your opponent's defending Pokémon has Unown G attached does nothing to protect you from the effects of that attack. Furthermore, even if you had Unown G attached to Garchomp C LV. X, you'd still have to discard energy and would not be able to use Dragon Rush the next turn, because Unown G does not protect you from effects of your own Pokémon's attacks.
I'd like to add that the effect prohibiting you from using Dragon Rush on the following turn would be removed if Garchomp were removed from the active position and then made active again (such as through a combination of retreating and Switch).
