QI play this follower's UB ability. During my opponent's next turn, can they play a Carbuncle's Sparkle targeting their Gilgamesh that hasn't attacked yet?

AYes, they can.

QI play this follower's UB ability. During my opponent's next turn, they play a Carbuncle's Sparkle with 0 cards in hand, returning a Gilgamesh that has already attacked to their hand. If they then summon the Gilgamesh, does it have to attack again before they can end their main phase?

AYes, it does.

QI play this follower's UB ability. During my opponent's next turn, they attack with an Arctic Chimera, which is then put into the cemetery as the result of combat or another card's effect, thereby triggering its Last Words ability. When it is put back onto the field, does it have to attack again before they can end their main phase?

AYes, it does.

QI play this follower's UB ability. During my opponent's next turn, they summon a Gilgamesh. Since it has Storm, does it have to attack before they can end their main phase, even though it was played after this follower's UB ability was played?

AYes. "While this is engaged, each enemy follower on the field must attack once per turn if able" is a passive ability, and as such applies even to enemy followers that weren't on the field when this follower's UB ability was activated.

QI play this follower's UB ability while my opponent has an evolved Assault Dragoon on the field. During their next turn, can they play its activated ability and engage it instead of attacking?

AYes, they can.

QI play this follower's UB ability while my opponent has an engaged Mejiro Palmer [Moonlit Devil ♪] on the field. At the start of their next turn's main phase, can they choose not to pay the cost of its automatic ability and leave it engaged?

AYes, they can.

QI play this follower's UB ability 2 times. During my opponent's next turn, they attack with a follower. If that follower is then refreshed, does it have to attack again before they can end their main phase?

ANo, it doesn't.