If this card becomes a copy of a resonator with Shift, it gains that Shift ability and may play the activate ability while it is in your field. However, this card does not copy the future side of a double-sided card, so the activate ability will resolve doing nothing.
If Shadow Doppelganger removes a resonator with a "when this card is put into a graveyard" automatic ability, it will trigger when Shadow Doppelganger goes to the graveyard from the field. If this ability returns Shadow Doppelganger to the field, the controlling player will be given the opportunity once again to remove a resonator from a graveyard.
This card has an "as this card enters your field" clause. This ability does not use the chase, and Shadow Doppelganger chooses a resonator to remove from the game as it resolves from the chase. The opposing player will not know which resonator the controlling player will remove until it is too late to respond.