yes you can equip spell card cards to an opponent's monster unless the card states so a good example of this is snatch steal
Yes. Cards like 'Snatch Steal' would be useless otherwise.
No, face-down monsters cannot be equipped.
No, the Equip card stays on your side of the field, and is under your control.
The equip spell is sent to the graveyard and the monster returns to your hand as usual :)