The benefits of equipping a monster to another monster varies, depending on the card. In the case of Union monsters, they usually have the ability to save your monster from destruction by sending themselves to the graveyard, and some have added bonuses, like "Heavy Mech Support Platform" adds 500 attack to your monster. "Dragunity" is an archetype built around equipping low level Dragon-type monsters to Winged Beast-type monsters, with various effects, but no protection effect. Other cards like "Allure Queen" and "Destiny Hero - Plasma" allow you to equip your opponent's monster(s) to them, thus removing potential threats from their field. Plasma also has the added benefit of gaining attack points equal to half the equipped monster's, and negating all of your opponent's monster effects.
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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.