Situation: Player A has {Izzet Staticaster} on the battlefield.
Player B casts {River's Grasp} and fulfills both mana conditions on the card (bounce and discard). {Izzet Staticaster} is targeted for the bounce effect.
Player A: taps the {Izzet Staticaster} in response deals 1 point of damage to a creature and kills it. Then returns the {Izzet Staticaster} to his hand.
At this point, Player A casts {Izzet Staticaster}, returning it to the Battlefield, before the discard effect resolves.
Is what Player A did legal? How should this have resolved?
Are the 2 effects of {River's Grasp} placed on the stack separately or do they occupy the same slot thus resolving at the same time?
Do the 2 effects have to resolve in the order as they appear on {River's Grasp}?
Player A misplayed.
{River's Grasp} resolves as one spell, with two effects. You don't get priority between the effects.
As to your second question: You resolve the effects in the order they are written on the card.
Take {Serum Visions} vs {Preordain}. In the first case, you'll draw, then scry. In the second, you'll scry first.
So, if I understand correctly, the {Izzet Staticaster} can't be targeted by the discard effect since it is on the battlefield when the {River's Grasp} has to choose targets for each effect.
No, it can be the card that gets discarded. The only targets for rivers grasp is the creature to be returned to hand, and the player that must reveal their hand and then discard.
The creature gets returned to its owners hand. Then player A reveals his hand and player B chooses which card he wants player A to discard.