Call me stupid, but can you equip to a face down, and.....can you flip a monster (like swarm of scarabs) face down after it has been equiped and keep the equipment????

No...you cannot equip a face-down monster...and no you cannot keep an equipment if the monster is flipped face-down or removed from the field in any way...basic game mechanics. If an Equip Card loses its target, it destroys itself and is sent to the Graveyard (or removed from play depending on effects).