Yes. Cards like 'Snatch Steal' would be useless otherwise.
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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
Yes, and in fact several Equip Spell cards would be useless if they could not, such as 'Snatch Steal' and 'Mark of the Rose'.
No, face-down monsters cannot be equipped.
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.