In a multiplayer game, if you leave the game after Mu Yanling's first ability resolves but before your next turn begins, its effect lasts until your next turn would have begun. It neither expires immediately nor lasts indefinitely.

Tapping an Island for mana is an activated ability. You can't activate that ability and also activate the ability it gains from Mu Yanling's emblem.

Mu Yanling's first ability can target a creature that doesn't have flying. The creature just gets -2/-0.

An Island is a land with the subtype Island, not any land with a mana ability that produces blue mana.

After Mu Yanling's first ability has resolved, the target creature will have flying again if another effect grants it flying.