Yes and no. If it says to for the cards effect or other card effect, then yes. Otherwise, no you must be given the ability to via card play.
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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.
I run a very good dragon deck and the only equip card I run is Axe of Despair. 1000 attack.
No, the Equip card stays on your side of the field, and is under your control.
No, face-down monsters cannot be equipped.
The Equip happens on resolution of an Ignition Effect. Book of Moon can be chained to that. The target monster will flip face-down and then the Equip effect will resolve without effect because it can't equip to a Set monster. Inzektor Hornet will not move from whatever zone it started in.