No, face-down monsters cannot be equipped.
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Same as when you use a regular Equip Spell card on the opponent's monster, the monster acting as an Equip Spell Card goes to your own S/T zone.
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
No, the Equip card stays on your side of the field, and is under your control.
Yes. Cards like 'Snatch Steal' would be useless otherwise.
There are 3 Spell Cards in the TCG that change a monster's type: the Equip Spell "Dragonic Attack", which changes the equipped monster into a Dragon; the Equip Spell "Reptillianne Rage", which changes the equipped monster into a Reptile; and the Field Spell "Zombie World" which changes all monsters in play and in the Graveyards into Zombies.