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
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Yes, and in fact several Equip Spell cards would be useless if they could not, such as 'Snatch Steal' and 'Mark of the Rose'.
Yes. Cards like 'Snatch Steal' would be useless otherwise.
Monsters who go into the S/T zone for any reason are then treated as Continuous Spell Cards or Equip Spell Cards depending on what it is doing. It is no longer a monster card so is unaffected by anything that specifically affects monster cards, and cannot be tributed for a tribute summon, etc. However if something required you to send an Equip Spell Card to the graveyard, you could use that monster.
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.
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.